# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a001147 Showing 1-1 of 1 %I A001147 M3002 N1217 #653 Nov 15 2024 23:32:27 %S A001147 1,1,3,15,105,945,10395,135135,2027025,34459425,654729075,13749310575, %T A001147 316234143225,7905853580625,213458046676875,6190283353629375, %U A001147 191898783962510625,6332659870762850625,221643095476699771875,8200794532637891559375,319830986772877770815625 %N A001147 Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1). %C A001147 The solution to Schröder's third problem. %C A001147 Number of fixed-point-free involutions in symmetric group S_{2n} (cf. A000085). %C A001147 a(n-2) is the number of full Steiner topologies on n points with n-2 Steiner points. [corrected by _Lyle Ramshaw_, Jul 20 2022] %C A001147 a(n) is also the number of perfect matchings in the complete graph K(2n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001 %C A001147 Number of ways to choose n disjoint pairs of items from 2*n items. - Ron Zeno (rzeno(AT)hotmail.com), Feb 06 2002 %C A001147 Number of ways to choose n-1 disjoint pairs of items from 2*n-1 items (one item remains unpaired). - _Bartosz Zoltak_, Oct 16 2012 %C A001147 For n >= 1 a(n) is the number of permutations in the symmetric group S_(2n) whose cycle decomposition is a product of n disjoint transpositions. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001 %C A001147 a(n) is the number of distinct products of n+1 variables with commutative, nonassociative multiplication. - Andrew Walters (awalters3(AT)yahoo.com), Jan 17 2004. For example, a(3)=15 because the product of the four variables w, x, y and z can be constructed in exactly 15 ways, assuming commutativity but not associativity: 1. w(x(yz)) 2. w(y(xz)) 3. w(z(xy)) 4. x(w(yz)) 5. x(y(wz)) 6. x(z(wy)) 7. y(w(xz)) 8. y(x(wz)) 9. y(z(wx)) 10. z(w(xy)) 11. z(x(wy)) 12. z(y(wx)) 13. (wx)(yz) 14. (wy)(xz) 15. (wz)(xy). %C A001147 a(n) = E(X^(2n)), where X is a standard normal random variable (i.e., X is normal with mean = 0, variance = 1). So for instance a(3) = E(X^6) = 15, etc. See Abramowitz and Stegun or Hoel, Port and Stone. - Jerome Coleman, Apr 06 2004 %C A001147 Second Eulerian transform of 1,1,1,1,1,1,... The second Eulerian transform transforms a sequence s to a sequence t by the formula t(n) = Sum_{k=0..n} E(n,k)s(k), where E(n,k) is a second-order Eulerian number (A008517). - _Ross La Haye_, Feb 13 2005 %C A001147 Integral representation as n-th moment of a positive function on the positive axis, in Maple notation: a(n) = int(x^n*exp(-x/2)/sqrt(2*Pi*x), x=0..infinity), n=0,1... . - _Karol A. Penson_, Oct 10 2005 %C A001147 a(n) is the number of binary total partitions of n+1 (each non-singleton block must be partitioned into exactly two blocks) or, equivalently, the number of unordered full binary trees with n+1 labeled leaves (Stanley, ex 5.2.6). - _Mitch Harris_, Aug 01 2006 %C A001147 a(n) is the Pfaffian of the skew-symmetric 2n X 2n matrix whose (i,j) entry is i for i=0} k*A079267(n,k). %C A001147 (End) %C A001147 Hankel transform is A137592. - _Paul Barry_, Sep 18 2009 %C A001147 (1, 3, 15, 105, ...) = INVERT transform of A000698 starting (1, 2, 10, 74, ...). - _Gary W. Adamson_, Oct 21 2009 %C A001147 a(n) = (-1)^(n+1)*H(2*n,0), where H(n,x) is the probabilists' Hermite polynomial. The generating function for the probabilists' Hermite polynomials is as follows: exp(x*t-t^2/2) = Sum_{i>=0} H(i,x)*t^i/i!. - _Leonid Bedratyuk_, Oct 31 2009 %C A001147 The Hankel transform of a(n+1) is A168467. - _Paul Barry_, Dec 04 2009 %C A001147 Partial products of odd numbers. - _Juri-Stepan Gerasimov_, Oct 17 2010 %C A001147 See A094638 for connections to differential operators. - _Tom Copeland_, Sep 20 2011 %C A001147 a(n) is the number of subsets of {1,...,n^2} that contain exactly k elements from {1,...,k^2} for k=1,...,n. For example, a(3)=15 since there are 15 subsets of {1,2,...,9} that satisfy the conditions, namely, {1,2,5}, {1,2,6}, {1,2,7}, {1,2,8}, {1,2,9}, {1,3,5}, {1,3,6}, {1,3,7}, {1,3,8}, {1,3,9}, {1,4,5}, {1,4,6}, {1,4,7}, {1,4,8}, and {1,4,9}. - _Dennis P. Walsh_, Dec 02 2011 %C A001147 a(n) is the leading coefficient of the Bessel polynomial y_n(x) (cf. A001498). - _Leonid Bedratyuk_, Jun 01 2012 %C A001147 For n>0: a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = min(i,j)^2 for 1 <= i,j <= n. - _Enrique Pérez Herrero_, Jan 14 2013 %C A001147 a(n) is also the numerator of the mean value from 0 to Pi/2 of sin(x)^(2n). - _Jean-François Alcover_, Jun 13 2013 %C A001147 a(n) is the size of the Brauer monoid on 2n points (see A227545). - _James Mitchell_, Jul 28 2013 %C A001147 For n>1: a(n) is the numerator of M(n)/M(1) where the numbers M(i) have the property that M(n+1)/M(n) ~ n-1/2 (for example, large Kendell-Mann numbers, see A000140 or A181609, as n --> infinity). - _Mikhail Gaichenkov_, Jan 14 2014 %C A001147 a(n) = the number of upper-triangular matrix representations required for the symbolic representation of a first order central moment of the multivariate normal distribution of dimension 2(n-1), i.e., E[X_1*X_2...*X_(2n-2)|mu=0, Sigma]. See vignette for symmoments R package on CRAN and Phillips reference below. - _Kem Phillips_, Aug 10 2014 %C A001147 For n>1: a(n) is the number of Feynman diagrams of order 2n (number of internal vertices) for the vacuum polarization with one charged loop only, in quantum electrodynamics. - _Robert Coquereaux_, Sep 15 2014 %C A001147 Aerated with intervening zeros (1,0,1,0,3,...) = a(n) (cf. A123023), the e.g.f. is e^(t^2/2), so this is the base for the Appell sequence A099174 with e.g.f. e^(t^2/2) e^(x*t) = exp(P(.,x),t) = unsigned A066325(x,t), the probabilist's (or normalized) Hermite polynomials. P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for A066325(x,t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), where UP(n,t) are the polynomials of A066325 and, e.g., (P(.,t))^n = P(n,t). - _Tom Copeland_, Nov 15 2014 %C A001147 a(n) = the number of relaxed compacted binary trees of right height at most one of size n. A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. The number of unbounded relaxed compacted binary trees of size n is A082161(n). See the Genitrini et al. link. - _Michael Wallner_, Jun 20 2017 %C A001147 Also the number of distinct adjacency matrices in the n-ladder rung graph. - _Eric W. Weisstein_, Jul 22 2017 %C A001147 From _Christopher J. Smyth_, Jan 26 2018: (Start) %C A001147 a(n) = the number of essentially different ways of writing a probability distribution taking n+1 values as a sum of products of binary probability distributions. See comment of Mitch Harris above. This is because each such way corresponds to a full binary tree with n+1 leaves, with the leaves labeled by the values. (This comment is due to Niko Brummer.) %C A001147 Also the number of binary trees with root labeled by an (n+1)-set S, its n+1 leaves by the singleton subsets of S, and other nodes labeled by subsets T of S so that the two daughter nodes of the node labeled by T are labeled by the two parts of a 2-partition of T. This also follows from Mitch Harris' comment above, since the leaf labels determine the labels of the other vertices of the tree. %C A001147 (End) %C A001147 a(n) is the n-th moment of the chi-squared distribution with one degree of freedom (equivalent to Coleman's Apr 06 2004 comment). - _Bryan R. Gillespie_, Mar 07 2021 %C A001147 Let b(n) = 0 for n odd and b(2k) = a(k); i.e., let the sequence b(n) be an aerated version of this entry. After expanding the differential operator (x + D)^n and normal ordering the resulting terms, the integer coefficient of the term x^k D^m is n! b(n-k-m) / [(n-k-m)! k! m!] with 0 <= k,m <= n and (k+m) <= n. E.g., (x+D)^2 = x^2 + 2xD + D^2 + 1 with D = d/dx. The result generalizes to the raising (R) and lowering (L) operators of any Sheffer polynomial sequence by replacing x by R and D by L and follows from the disentangling relation e^{t(L+R)} = e^{t^2/2} e^{tR} e^{tL}. Consequently, these are also the coefficients of the reordered 2^n permutations of the binary symbols L and R under the condition LR = RL + 1. E.g., (L+R)^2 = LL + LR + RL + RR = LL + 2RL + RR + 1. (Cf. A344678.) - _Tom Copeland_, May 25 2021 %C A001147 From _Tom Copeland_, Jun 14 2021: (Start) %C A001147 Lando and Zvonkin present several scenarios in which the double factorials occur in their role of enumerating perfect matchings (pairings) and as the nonzero moments of the Gaussian e^(x^2/2). %C A001147 Speyer and Sturmfels (p. 6) state that the number of facets of the abstract simplicial complex known as the tropical Grassmannian G'''(2,n), the space of phylogenetic T_n trees (see A134991), or Whitehouse complex is a shifted double factorial. %C A001147 These are also the unsigned coefficients of the x[2]^m terms in the partition polynomials of A134685 for compositional inversion of e.g.f.s, a refinement of A134991. %C A001147 a(n)*2^n = A001813(n) and A001813(n)/(n+1)! = A000108(n), the Catalan numbers, the unsigned coefficients of the x[2]^m terms in the partition polynomials A133437 for compositional inversion of o.g.f.s, a refinement of A033282, A126216, and A086810. Then the double factorials inherit a multitude of analytic and combinatoric interpretations from those of the Catalan numbers, associahedra, and the noncrossing partitions of A134264 with the Catalan numbers as unsigned-row sums. (End) %C A001147 Connections among the Catalan numbers A000108, the odd double factorials, values of the Riemann zeta function and its derivative for integer arguments, and series expansions of the reduced action for the simple harmonic oscillator and the arc length of the spiral of Archimedes are given in the MathOverflow post on the Riemann zeta function. - _Tom Copeland_, Oct 02 2021 %C A001147 b(n) = a(n) / (n! 2^n) = Sum_{k = 0..n} (-1)^n binomial(n,k) (-1)^k a(k) / (k! 2^k) = (1-b.)^n, umbrally; i.e., the normalized double factorial a(n) is self-inverse under the binomial transform. This can be proved by applying the Euler binomial transformation for o.g.f.s Sum_{n >= 0} (1-b_n)^n x^n = (1/(1-x)) Sum_{n >= 0} b_n (x / (x-1))^n to the o.g.f. (1-x)^{-1/2} = Sum_{n >= 0} b_n x^n. Other proofs are suggested by the discussion in Watson on pages 104-5 of transformations of the Bessel functions of the first kind with b(n) = (-1)^n binomial(-1/2,n) = binomial(n-1/2,n) = (2n)! / (n! 2^n)^2. - _Tom Copeland_, Dec 10 2022 %D A001147 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, (26.2.28). %D A001147 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 317. %D A001147 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 228, #19. %D A001147 Hoel, Port and Stone, Introduction to Probability Theory, Section 7.3. %D A001147 F. K. Hwang, D. S. Richards and P. Winter, The Steiner Tree Problem, North-Holland, 1992, see p. 14. %D A001147 C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980, pages 466-467. %D A001147 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A001147 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A001147 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.6 and also p. 178. %D A001147 R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer-Verlag, New York, 1999, p. 73. %D A001147 G. Watson, The Theory of Bessel Functions, Cambridge Univ. Press, 1922. %H A001147 Stefano Spezia, Table of n, a(n) for n = 0..400 (first 102 terms from T. D. Noe) %H A001147 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. %H A001147 José A. Adell and Beáta Bényi, Probabilistic Stirling numbers and applications, Aequat. Math. (2024). See p. 18. %H A001147 Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts). %H A001147 Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443. %H A001147 D. Arques and J.-F. Beraud, Rooted maps on orientable surfaces, Riccati's equation and continued fractions, Discrete Math., 215 (2000), 1-12. %H A001147 Fatemeh Bagherzadeh, M. Bremner, and S. Madariaga, Jordan Trialgebras and Post-Jordan Algebras, arXiv preprint arXiv:1611.01214 [math.RA], 2016. %H A001147 Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018. %H A001147 Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5. %H A001147 Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018. %H A001147 Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2. %H A001147 Natasha Blitvić and Einar Steingrímsson, Permutations, moments, measures, arXiv:2001.00280 [math.CO], 2020. %H A001147 O. Bodini, M. Dien, X. Fontaine, A. Genitrini, and H. K. Hwang, Increasing Diamonds, in LATIN 2016: 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings Pages pp. 207-219 2016 DOI 10.1007/978-3-662-49529-2_16; Lecture Notes in Computer Science Series Volume 9644. %H A001147 H. Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311 %H A001147 Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, May 23, 2011. %H A001147 David Callan, A combinatorial survey of identities for the double factorial, arXiv:0906.1317 [math.CO], 2009. %H A001147 Peter J. Cameron, Some treelike objects Quart. J. Math. Oxford Ser. 38 (1987), 155-183. MR0891613 (89a:05009). See p. 155. %H A001147 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A001147 Carsten Schneider, The Absent-Minded Passengers Problem: A Motivating Challenge Solved by Computer Algebra, arXiv:2003.01921 [math.CO], 2020. %H A001147 P. Codara, O. M. D'Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013. %H A001147 Thierry Dana-Picard, Sequences of Definite Integrals, Factorials and Double Factorials, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.6. %H A001147 Filippo Disanto and Thomas Wiehe, Some combinatorial problems on binary rooted trees occurring in population genetics, arXiv preprint arXiv:1112.1295 [math.CO], 2011. %H A001147 John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 5. %H A001147 J. Felsenstein, The number of evolutionary trees, Systematic Zoology, 27 (1978), 27-33. (Annotated scanned copy) %H A001147 FindStat - Combinatorial Statistic Finder, Perfect matchings %H A001147 A. Genitrini, B. Gittenberger, M. Kauers and M. Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017. %H A001147 Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1. %H A001147 S. Goodenough and C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv:1404.1894 [cs.DM], 2014. %H A001147 S. Goodenough and C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16. %H A001147 W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 Mar 2013. %H A001147 Paul W. Haggard, On Legendre numbers, International Journal of Mathematics and Mathematical Sciences, vol. 8, Article ID 787189, 5 pages, 1985. See Table 1 p. 408. %H A001147 Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy] %H A001147 Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020. %H A001147 Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011. %H A001147 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 23. %H A001147 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 106. %H A001147 J. Jakes-Schauer, D. Anekstein, and P. Wocjan, Carving-width and contraction trees for tensor networks, arXiv:1908.11034 [cs.DM], 2019. %H A001147 L. B. W. Jolley, Summation of Series, Dover, 1961 p. 48. %H A001147 M. Kauers and S.-L. Ko, Problem 11545, Amer. Math. Monthly, 118 (2011), p. 84. %H A001147 A. Khruzin, Enumeration of chord diagrams, arXiv:math/0008209 [math.CO], 2000. %H A001147 M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740. %H A001147 S. Lando and A. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, 141, Springer, 2004. %H A001147 Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. %H A001147 F. Larrion, M. A. Pizana, and R. Villarroel-Flores, The clique operator on matching and chessboard graphs Discrete Math. 309 (2009), no. 1, 85-93. %H A001147 Peter D. Loly and Ian D. Cameron, Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^L, L = 1, 2, 3, ..., with Information Entropy, arXiv:2008.11020 [math.HO], 2020. %H A001147 E. Lucas, Theorie des nombres (annotated scans of a few selected pages). %H A001147 Robert J. Marsh and Paul Martin, Tiling bijections between paths and Brauer diagrams, Journal of Algebraic Combinatorics, Vol 33, No 3 (2011), pp. 427-453. %H A001147 MathOverflow, Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?, answer by Tom Copeland posted in Aug 2021. %H A001147 B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425-426. %H A001147 T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984. %H A001147 T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360. %H A001147 F. Murtagh, Counting dendrograms: a survey, Discrete Applied Mathematics, 7 (1984), 191-199. %H A001147 G. Nordh, Perfect Skolem sequences, arXiv:math/0506155 [math.CO], 2005. %H A001147 J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. %H A001147 R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231. %H A001147 L. Pachter and B. Sturmfels, The mathematics of phylogenomics, arXiv:math/0409132 [math.ST], 2004-2005. %H A001147 K. Phillips, R functions to symbolically compute the central moments of the multivariate normal distribution, Journal of Statistical Software, Feb 2010. %H A001147 R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007. %H A001147 Helmut Prodinger, Descendants in heap ordered trees or a triumph of computer algebra, The Electronic Journal of Combinatorics, Volume 3, Issue 1 (1996), R29. %H A001147 S. Ramanujan, Question 541, J. Ind. Math. Soc. %H A001147 D. F. Robinson, Comparison of labeled trees with valency three, J. Combin. Theory Ser. B, 11 (1971), 105-119. %H A001147 M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), 10.6.7, (6.27). %H A001147 E. Schröder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376. %H A001147 E. Schröder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376. [Annotated scanned copy] %H A001147 Y. S. Song, On the combinatorics of rooted binary phylogenetic trees, Annals of Combinatorics, 7, 2003, 365-379. See Lemma 2.1. - _N. J. A. Sloane_, Aug 22 2014 %H A001147 D. Speyer and B. Sturmfels, The tropical Grassmannian, arXiv:0304218 [math.AG], 2003. %H A001147 Neriman Tokcan, Jonathan Gryak, Kayvan Najarian, and Harm Derksen, Algebraic Methods for Tensor Data, arXiv:2005.12988 [math.RT], 2020. %H A001147 Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019). %H A001147 Andrew Vince and Miklos Bona, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012. %H A001147 Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017 %H A001147 Eric Weisstein's World of Mathematics, Adjacency Matrix %H A001147 Eric Weisstein's World of Mathematics, Double Factorial %H A001147 Eric Weisstein's World of Mathematics, Erf %H A001147 Eric Weisstein's World of Mathematics, Ladder Rung Graph %H A001147 Eric Weisstein's World of Mathematics, Normal Distribution Function %H A001147 Wikipedia, Pfaffian %H A001147 Wikipedia, Hermite polynomials %H A001147 Index to divisibility sequences %H A001147 Index entries for related partition-counting sequences %H A001147 Index entries for sequences related to factorial numbers %H A001147 Index entries for sequences related to parenthesizing %H A001147 Index entries for "core" sequences %F A001147 E.g.f.: 1 / sqrt(1 - 2*x). %F A001147 D-finite with recurrence: a(n) = a(n-1)*(2*n-1) = (2*n)!/(n!*2^n) = A010050(n)/A000165(n). %F A001147 a(n) ~ sqrt(2) * 2^n * (n/e)^n. %F A001147 Rational part of numerator of Gamma(n+1/2): a(n) * sqrt(Pi) / 2^n = Gamma(n+1/2). - Yuriy Brun, Ewa Dominowska (brun(AT)mit.edu), May 12 2001 %F A001147 With interpolated zeros, the sequence has e.g.f. exp(x^2/2). - _Paul Barry_, Jun 27 2003 %F A001147 The Ramanujan polynomial psi(n+1, n) has value a(n). - _Ralf Stephan_, Apr 16 2004 %F A001147 a(n) = Sum_{k=0..n} (-2)^(n-k)*A048994(n, k). - _Philippe Deléham_, Oct 29 2005 %F A001147 Log(1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + ...) = x + 5/2*x^2 + 37/3*x^3 + 353/4*x^4 + 4081/5*x^5 + 55205/6*x^6 + ..., where [1, 5, 37, 353, 4081, 55205, ...] = A004208. - _Philippe Deléham_, Jun 20 2006 %F A001147 1/3 + 2/15 + 3/105 + ... = 1/2. [Jolley eq. 216] %F A001147 Sum_{j=1..n} j/a(j+1) = (1 - 1/a(n+1))/2. [Jolley eq. 216] %F A001147 1/1 + 1/3 + 2/15 + 6/105 + 24/945 + ... = Pi/2. - _Gary W. Adamson_, Dec 21 2006 %F A001147 a(n) = (1/sqrt(2*Pi))*Integral_{x>=0} x^n*exp(-x/2)/sqrt(x). - _Paul Barry_, Jan 28 2008 %F A001147 a(n) = A006882(2n-1). - _R. J. Mathar_, Jul 04 2009 %F A001147 G.f.: 1/(1-x-2x^2/(1-5x-12x^2/(1-9x-30x^2/(1-13x-56x^2/(1- ... (continued fraction). - _Paul Barry_, Sep 18 2009 %F A001147 a(n) = (-1)^n*subs({log(e)=1,x=0},coeff(simplify(series(e^(x*t-t^2/2),t,2*n+1)),t^(2*n))*(2*n)!). - _Leonid Bedratyuk_, Oct 31 2009 %F A001147 a(n) = 2^n*gamma(n+1/2)/gamma(1/2). - _Jaume Oliver Lafont_, Nov 09 2009 %F A001147 G.f.: 1/(1-x/(1-2x/(1-3x/(1-4x/(1-5x/(1- ...(continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009 %F A001147 The g.f. of a(n+1) is 1/(1-3x/(1-2x/(1-5x/(1-4x/(1-7x/(1-6x/(1-.... (continued fraction). - _Paul Barry_, Dec 04 2009 %F A001147 a(n) = Sum_{i=1..n} binomial(n,i)*a(i-1)*a(n-i). - _Vladimir Shevelev_, Sep 30 2010 %F A001147 E.g.f.: A(x) = 1 - sqrt(1-2*x) satisfies the differential equation A'(x) - A'(x)*A(x) - 1 = 0. - _Vladimir Kruchinin_, Jan 17 2011 %F A001147 a(n) = A123023(2*n + 1). - _Michael Somos_, Jul 24 2011 %F A001147 a(n) = (1/2)*Sum_{i=1..n} binomial(n+1,i)*a(i-1)*a(n-i). See link above. - _Dennis P. Walsh_, Dec 02 2011 %F A001147 a(n) = Sum_{k=0..n} (-1)^k*binomial(2*n,n+k)*Stirling_1(n+k,k) [Kauers and Ko]. %F A001147 a(n) = A035342(n, 1), n >= 1 (first column of triangle). %F A001147 a(n) = A001497(n, 0) = A001498(n, n), first column, resp. main diagonal, of Bessel triangle. %F A001147 From _Gary W. Adamson_, Jul 19 2011: (Start) %F A001147 a(n) = upper left term of M^n and sum of top row terms of M^(n-1), where M = a variant of the (1,2) Pascal triangle (Cf. A029635) as the following production matrix: %F A001147 1, 2, 0, 0, 0, ... %F A001147 1, 3, 2, 0, 0, ... %F A001147 1, 4, 5, 2, 0, ... %F A001147 1, 5, 9, 7, 2, ... %F A001147 ... %F A001147 For example, a(3) = 15 is the left term in top row of M^3: (15, 46, 36, 8) and a(4) = 105 = (15 + 46 + 36 + 8). %F A001147 (End) %F A001147 G.f.: A(x) = 1 + x/(W(0) - x); W(k) = 1 + x + x*2*k - x*(2*k + 3)/W(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 17 2011 %F A001147 a(n) = Sum_{i=1..n} binomial(n,i-1)*a(i-1)*a(n-i). - _Dennis P. Walsh_, Dec 02 2011 %F A001147 a(n) = A009445(n) / A014481(n). - _Reinhard Zumkeller_, Dec 03 2011 %F A001147 a(n) = (-1)^n*Sum_{k=0..n} 2^(n-k)*s(n+1,k+1), where s(n,k) are the Stirling numbers of the first kind, A048994. - _Mircea Merca_, May 03 2012 %F A001147 a(n) = (2*n)_4! = Gauss_factorial(2*n,4) = Product_{j=1..2*n, gcd(j,4)=1} j. - _Peter Luschny_, Oct 01 2012 %F A001147 G.f.: (1 - 1/Q(0))/x where Q(k) = 1 - x*(2*k - 1)/(1 - x*(2*k + 2)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 19 2013 %F A001147 G.f.: 1 + x/Q(0), where Q(k) = 1 + (2*k - 1)*x - 2*x*(k + 1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 01 2013 %F A001147 G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k + 1)/(2*x*(2*k + 1) - 1 + 2*x*(2*k + 2)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 31 2013 %F A001147 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(2*k + 1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 01 2013 %F A001147 G.f.: G(0), where G(k) = 1 + 2*x*(4*k + 1)/(4*k + 2 - 2*x*(2*k + 1)*(4*k + 3)/(x*(4*k + 3) + 2*(k + 1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 22 2013 %F A001147 a(n) = (2*n - 3)*a(n-2) + (2*n - 2)*a(n-1), n > 1. - _Ivan N. Ianakiev_, Jul 08 2013 %F A001147 G.f.: G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Aug 04 2013 %F A001147 a(n) = 2*a(n-1) + (2n-3)^2*a(n-2), a(0) = a(1) = 1. - _Philippe Deléham_, Oct 27 2013 %F A001147 G.f. of reciprocals: Sum_{n>=0} x^n/a(n) = 1F1(1; 1/2; x/2), confluent hypergeometric Function. - _R. J. Mathar_, Jul 25 2014 %F A001147 0 = a(n)*(+2*a(n+1) - a(n+2)) + a(n+1)*(+a(n+1)) for all n in Z. - _Michael Somos_, Sep 18 2014 %F A001147 a(n) = (-1)^n / a(-n) = 2*a(n-1) + a(n-1)^2 / a(n-2) for all n in Z. - _Michael Somos_, Sep 18 2014 %F A001147 From _Peter Bala_, Feb 18 2015: (Start) %F A001147 Recurrence equation: a(n) = (3*n - 2)*a(n-1) - (n - 1)*(2*n - 3)*a(n-2) with a(1) = 1 and a(2) = 3. %F A001147 The sequence b(n) = A087547(n), beginning [1, 4, 52, 608, 12624, ... ], satisfies the same second-order recurrence equation. This leads to the generalized continued fraction expansion lim_{n -> infinity} b(n)/a(n) = Pi/2 = 1 + 1/(3 - 6/(7 - 15/(10 - ... - n*(2*n - 1)/((3*n + 1) - ... )))). (End) %F A001147 E.g.f of the sequence whose n-th element (n = 1,2,...) equals a(n-1) is 1-sqrt(1-2*x). - _Stanislav Sykora_, Jan 06 2017 %F A001147 Sum_{n >= 1} a(n)/(2*n-1)! = exp(1/2). - _Daniel Suteu_, Feb 06 2017 %F A001147 a(n) = A028338(n, 0), n >= 0. - _Wolfdieter Lang_, May 27 2017 %F A001147 a(n) = (Product_{k=0..n-2} binomial(2*(n-k),2))/n!. - _Stefano Spezia_, Nov 13 2018 %F A001147 a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} C(n-1,i)*C(n-i-1,j)*a(i)*a(j)*a(n-i-j-1), a(0)=1, - _Vladimir Kruchinin_, May 06 2020 %F A001147 From _Amiram Eldar_, Jun 29 2020: (Start) %F A001147 Sum_{n>=1} 1/a(n) = sqrt(e*Pi/2)*erf(1/sqrt(2)), where erf is the error function. %F A001147 Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(Pi/(2*e))*erfi(1/sqrt(2)), where erfi is the imaginary error function. (End) %F A001147 G.f. of reciprocals: R(x) = Sum_{n>=0} x^n/a(n) satisfies (1 + x)*R(x) = 1 + 2*x*R'(x). - _Werner Schulte_, Nov 04 2024 %e A001147 a(3) = 1*3*5 = 15. %e A001147 From _Joerg Arndt_, Sep 10 2013: (Start) %e A001147 There are a(3)=15 involutions of 6 elements without fixed points: %e A001147 #: permutation transpositions %e A001147 01: [ 1 0 3 2 5 4 ] (0, 1) (2, 3) (4, 5) %e A001147 02: [ 1 0 4 5 2 3 ] (0, 1) (2, 4) (3, 5) %e A001147 03: [ 1 0 5 4 3 2 ] (0, 1) (2, 5) (3, 4) %e A001147 04: [ 2 3 0 1 5 4 ] (0, 2) (1, 3) (4, 5) %e A001147 05: [ 2 4 0 5 1 3 ] (0, 2) (1, 4) (3, 5) %e A001147 06: [ 2 5 0 4 3 1 ] (0, 2) (1, 5) (3, 4) %e A001147 07: [ 3 2 1 0 5 4 ] (0, 3) (1, 2) (4, 5) %e A001147 08: [ 3 4 5 0 1 2 ] (0, 3) (1, 4) (2, 5) %e A001147 09: [ 3 5 4 0 2 1 ] (0, 3) (1, 5) (2, 4) %e A001147 10: [ 4 2 1 5 0 3 ] (0, 4) (1, 2) (3, 5) %e A001147 11: [ 4 3 5 1 0 2 ] (0, 4) (1, 3) (2, 5) %e A001147 12: [ 4 5 3 2 0 1 ] (0, 4) (1, 5) (2, 3) %e A001147 13: [ 5 2 1 4 3 0 ] (0, 5) (1, 2) (3, 4) %e A001147 14: [ 5 3 4 1 2 0 ] (0, 5) (1, 3) (2, 4) %e A001147 15: [ 5 4 3 2 1 0 ] (0, 5) (1, 4) (2, 3) %e A001147 (End) %e A001147 G.f. = 1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + 135135*x^7 + ... %p A001147 f := n->(2*n)!/(n!*2^n); %p A001147 A001147 := proc(n) doublefactorial(2*n-1); end: # _R. J. Mathar_, Jul 04 2009 %p A001147 A001147 := n -> 2^n*pochhammer(1/2, n); # _Peter Luschny_, Aug 09 2009 %p A001147 G(x):=(1-2*x)^(-1/2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..19); # _Zerinvary Lajos_, Apr 03 2009; aligned with offset by _Johannes W. Meijer_, Aug 11 2009 %p A001147 series(hypergeom([1,1/2],[],2*x),x=0,20); # _Mark van Hoeij_, Apr 07 2013 %t A001147 Table[(2 n - 1)!!, {n, 0, 19}] (* _Robert G. Wilson v_, Oct 12 2005 *) %t A001147 a[ n_] := 2^n Gamma[n + 1/2] / Gamma[1/2]; (* _Michael Somos_, Sep 18 2014 *) %t A001147 Join[{1}, Range[1, 41, 2]!!] (* _Harvey P. Dale_, Jan 28 2017 *) %t A001147 a[ n_] := If[ n < 0, (-1)^n / a[-n], SeriesCoefficient[ Product[1 - (1 - x)^(2 k - 1), {k, n}], {x, 0, n}]]; (* _Michael Somos_, Jun 27 2017 *) %t A001147 (2 Range[0, 20] - 1)!! (* _Eric W. Weisstein_, Jul 22 2017 *) %o A001147 (PARI) {a(n) = if( n<0, (-1)^n / a(-n), (2*n)! / n! / 2^n)}; /* _Michael Somos_, Sep 18 2014 */ %o A001147 (PARI) x='x+O('x^33); Vec(serlaplace((1-2*x)^(-1/2))) \\ _Joerg Arndt_, Apr 24 2011 %o A001147 (Magma) A001147:=func< n | n eq 0 select 1 else &*[ k: k in [1..2*n-1 by 2] ] >; [ A001147(n): n in [0..20] ]; // _Klaus Brockhaus_, Jun 22 2011 %o A001147 (Magma) I:=[1,3]; [1] cat [n le 2 select I[n] else (3*n-2)*Self(n-1)-(n-1)*(2*n-3)*Self(n-2): n in [1..25] ]; // _Vincenzo Librandi_, Feb 19 2015 %o A001147 (Haskell) %o A001147 a001147 n = product [1, 3 .. 2 * n - 1] %o A001147 a001147_list = 1 : zipWith (*) [1, 3 ..] a001147_list %o A001147 -- _Reinhard Zumkeller_, Feb 15 2015, Dec 03 2011 %o A001147 (Sage) [rising_factorial(n+1,n)/2^n for n in (0..15)] # _Peter Luschny_, Jun 26 2012 %o A001147 (Python) %o A001147 from sympy import factorial2 %o A001147 def a(n): return factorial2(2 * n - 1) %o A001147 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jul 22 2017 %o A001147 (GAP) A001147 := function(n) local i, s, t; t := 1; i := 0; Print(t, ", "); for i in [1 .. n] do t := t*(2*i-1); Print(t, ", "); od; end; A001147(100); # _Stefano Spezia_, Nov 13 2018 %o A001147 (Maxima) %o A001147 a(n):=if n=0 then 1 else sum(sum(binomial(n-1,i)*binomial(n-i-1,j)*a(i)*a(j)*a(n-i-j-1),j,0,n-i-1),i,0,n-1); /* _Vladimir Kruchinin_, May 06 2020 */ %Y A001147 Cf. A000085, A006882, A000165 ((2n)!!), A001818, A009445, A039683, A102992, A001190 (no labels), A000680, A132101. %Y A001147 Cf. A086677; A055142 (for this sequence, |a(n+1)| + 1 is the number of distinct products which can be formed using commutative, nonassociative multiplication and a nonempty subset of n given variables). %Y A001147 Constant terms of polynomials in A098503. First row of array A099020. %Y A001147 Cf. A079267, A000698, A029635, A161198, A076795, A123023, A161124, A051125, A181983, A099174, A087547, A028338 (first column). %Y A001147 Subsequence of A248652. %Y A001147 Cf. A082161 (relaxed compacted binary trees of unbounded right height). %Y A001147 Cf. A053871 (binomial transform). %Y A001147 Cf. A000108, A001813, A033282, A060540, A086810, A094638, A126216, A133437, A134264, A134685, A134991, A344678. %K A001147 nonn,easy,nice,core,changed %O A001147 0,3 %A A001147 _N. J. A. Sloane_ %E A001147 Removed erroneous comments: neither the number of n X n binary matrices A such that A^2 = 0 nor the number of simple directed graphs on n vertices with no directed path of length two are counted by this sequence (for n = 3, both are 13). - _Dan Drake_, Jun 02 2009 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE