%I #189 Aug 28 2024 04:58:59

%S 1,1,1,1,3,2,1,6,11,6,1,10,35,50,24,1,15,85,225,274,120,1,21,175,735,

%T 1624,1764,720,1,28,322,1960,6769,13132,13068,5040,1,36,546,4536,

%U 22449,67284,118124,109584,40320,1,45,870,9450,63273,269325,723680,1172700,1026576,362880

%N Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).

%C Triangle of coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in decreasing powers of x. - _T. D. Noe_, Feb 22 2008

%C Row n also gives the number of permutation of 1..n with complexity 0,1,...,n-1. See the comments in A008275. - _N. J. A. Sloane_, Feb 08 2019

%C T(n,k) is the number of deco polyominoes of height n and having k columns. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: T(2,1)=1 and T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 1 and 2 columns. - _Emeric Deutsch_, Aug 14 2006

%C Sum_{k=1..n} k*T(n,k) = A121586. - _Emeric Deutsch_, Aug 14 2006

%C Let the triangle U(n,k), 0 <= k <= n, read by rows, be given by [1,0,1,0,1,0,1,0,1,0,1,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,...] where DELTA is the operator defined in A084938; then T(n,k) = U(n-1,k-1). - _Philippe Deléham_, Jan 06 2007

%C From _Tom Copeland_, Dec 15 2007: (Start)

%C Consider c(t) = column vector(1, t, t^2, t^3, t^4, t^5, ...).

%C Starting at 1 and sampling every integer to the right, we obtain (1,2,3,4,5,...). And T * c(1) = (1, 1*2, 1*2*3, 1*2*3*4,...), giving n! for n > 0. Call this sequence the right factorial (n+)!.

%C Starting at 1 and sampling every integer to the left, we obtain (1,0,-1,-2,-3,-4,-5,...). And T * c(-1) = (1, 1*0, 1*0*-1, 1*0*-1*-2,...) = (1, 0, 0, 0, ...), the left factorial (n-)!.

%C Sampling every other integer to the right, we obtain (1,3,5,7,9,...). T * c(2) = (1, 1*3, 1*3*5, ...) = (1,3,15,105,945,...), giving A001147 for n > 0, the right double factorial, (n+)!!.

%C Sampling every other integer to the left, we obtain (1,-1,-3,-5,-7,...). T * c(-2) = (1, 1*-1, 1*-1*-3, 1*-1*-3*-5,...) = (1,-1,3,-15,105,-945,...) = signed A001147, the left double factorial, (n-)!!.

%C Sampling every 3 steps to the right, we obtain (1,4,7,10,...). T * c(3) = (1, 1*4, 1*4*7,...) = (1,4,28,280,...), giving A007559 for n > 0, the right triple factorial, (n+)!!!.

%C Sampling every 3 steps to the left, we obtain (1,-2,-5,-8,-11,...), giving T * c(-3) = (1, 1*-2, 1*-2*-5, 1*-2*-5*-8,...) = (1,-2,10,-80,880,...) = signed A008544, the left triple factorial, (n-)!!!.

%C The list partition transform A133314 of [1,T * c(t)] gives [1,T * c(-t)] with all odd terms negated; e.g., LPT[1,T*c(2)] = (1,-1,-1,-3,-15,-105,-945,...) = (1,-A001147). And e.g.f. for [1,T * c(t)] = (1-xt)^(-1/t).

%C The above results hold for t any real or complex number. (End)

%C Let R_n(x) be the real and I_n(x) the imaginary part of Product_{k=0..n} (x + I*k). Then, for n=1,2,..., we have R_n(x) = Sum_{k=0..floor((n+1)/2)}(-1)^k*Stirling1(n+1,n+1-2*k)*x^(n+1-2*k), I_n(x) = Sum_{k=0..floor(n/2)}(-1)^(k+1)*Stirling1(n+1,n-2*k)*x^(n-2*k). - _Milan Janjic_, May 11 2008

%C T(n,k) is also the number of permutations of n with "reflection length" k (i.e., obtained from 12..n by k not necessarily adjacent transpositions). For example, when n=3, 132, 213, 321 are obtained by one transposition, while 231 and 312 require two transpositions. - _Kyle Petersen_, Oct 15 2008

%C From _Tom Copeland_, Nov 02 2010: (Start)

%C [x^(y+1) D]^n = x^(n*y) [T(n,1)(xD)^n + T(n,2)y (xD)^(n-1) + ... + T(n,n)y^(n-1)(xD)], with D the derivative w.r.t. x.

%C E.g., [x^(y+1) D]^4 = x^(4*y) [(xD)^4 + 6 y(xD)^3 + 11 y^2(xD)^2 + 6 y^3(xD)].

%C (xD)^m can be further expanded in terms of the Stirling numbers of the second kind and operators of the form x^j D^j. (End)

%C With offset 0, 0 <= k <= n: T(n,k) is the sum of products of each size k subset of {1,2,...,n}. For example, T(3,2) = 11 because there are three subsets of size two: {1,2},{1,3},{2,3}. 1*2 + 1*3 + 2*3 = 11. - _Geoffrey Critzer_, Feb 04 2011

%C The Kn11, Fi1 and Fi2 triangle sums link this triangle with two sequences, see the crossrefs. For the definitions of these triangle sums see A180662. The mirror image of this triangle is A130534. - _Johannes W. Meijer_, Apr 20 2011

%C T(n+1,k+1) is the elementary symmetric function a_k(1,2,...,n), n >= 0, k >= 0, (a_0(0):=1). See the _T. D. Noe_ and _Geoffrey Critzer_ comments given above. For a proof see the Stanley reference, p. 19, Second Proof. - _Wolfdieter Lang_, Oct 24 2011

%C Let g(t) = 1/d(log(P(j+1,-t)))/dt (see _Tom Copeland_'s 2007 formulas). The Mellin transform (t to s) of t*Dirac[g(t)] gives Sum_{n=1..j} n^(-s), which as j tends to infinity gives the Riemann zeta function for Re(s) > 1. Dirac(x) is the Dirac delta function. The complex contour integral along a circle of radius 1 centered at z=1 of z^s/g(z) gives the same result. - _Tom Copeland_, Dec 02 2011

%C Rows are coefficients of the polynomial expansions of the Pochhammer symbol, or rising factorial, Pch(n,x) = (x+n-1)!/(x-1)!. Expansion of Pch(n,xD) = Pch(n,Bell(.,:xD:)) in a polynomial with terms :xD:^k=x^k*D^k gives the Lah numbers A008297. Bell(n,x) are the unsigned Bell polynomials or Stirling polynomials of the second kind A008277. - _Tom Copeland_, Mar 01 2014

%C From _Tom Copeland_, Dec 09 2016: (Start)

%C The Betti numbers, or dimension, of the pure braid group cohomology. See pp. 12 and 13 of the Hyde and Lagarias link.

%C Row polynomials and their products appear in presentation of the Jack symmetric functions of R. Stanley. See Copeland link on the Witt differential generator.

%C (End)

%C From _Tom Copeland_, Dec 16 2019: (Start)

%C The e.g.f. given by Copeland in the formula section appears in a combinatorial Dyson-Schwinger equation of quantum field theory in Yeats in Thm. 2 on p. 62 related to a Hopf algebra of rooted trees. See also the Green function on p. 70.

%C Per comments above, this array contains the coefficients in the expansion in polynomials of the Euler, or state number, operator xD of the rising factorials Pch(n,xD) = (xD+n-1)!/(xD-1)! = x [:Dx:^n/n!]x^{-1} = L_n^{-1}(-:xD:), where :Dx:^n = D^n x^n and :xD:^n = x^n D^n. The polynomials L_n^{-1} are the Laguerre polynomials of order -1, i.e., normalized Lah polynomials.

%C The Witt differential operators L_n = x^(n+1) D and the row e.g.f.s appear in Hopf and dual Hopf algebra relations presented by Foissy. The Witt operators satisfy L_n L_k - L_k L_n = (k-n) L_(n+k), as for the dual Hopf algebra. (End)

%D M. Miyata and J. W. Son, On the complexity of permutations and the metric space of bijections, Tensor, 60 (1998), No. 1, 109-116 (MR1768839).

%D R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.

%H T. D. Noe, <a href="/A094638/b094638.txt">Rows n = 1..51 of triangle, flattened</a>

%H E. Barcucci, A. Del Lungo and R. Pinzani, <a href="http://dx.doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.

%H F. Bergeron, Ph. Flajolet and B. Salvy, <a href="http://algo.inria.fr/flajolet/Publications/BeFlSa92.pdf">Varieties of Increasing Trees</a>, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48. Added Mar 01 2014

%H F. Bergeron, Philippe Flajolet and Bruno Salvy, <a href="http://hal.inria.fr/inria-00074977">Varieties of increasing trees</a>, HAL, Rapport De Recherche Inria. Added Mar 01 2014

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2010/12/28/14/">Addendum to Mathemagical Forests</a>

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/08/23/a-class-of-differential-operators-and-the-stirling-numbers/">A Class of Differential Operators and the Stirling Numbers</a>

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/09/16/mellin-transform-interpolation-of-differential-operators/">Mellin Interpolation of Differential Ops and Associated Infinigens and Appell Polynomials: The Ordered, Laguerre, and Scherk-Witt-Lie Diff Ops</a>

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2016/11/27/a-note-on-the-jack-symmetric-functions-polynomials/">Witt differential generator for special Jack symmetric functions / polynomials</a>

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000216/">The absolute length of a permutation</a>

%H L. Foissy, <a href="http://arXiv.org/abs/0707.1204">Faa di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations</a>, arXiv:0707.1204 [math.RA], (2007).

%H O. Furdui, T. Trif, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Furdui/furdui3.html">On the Summation of Certain Iterated Series</a>, J. Int. Seq. 14 (2011) #11.6.1

%H F. Hivert, J.-C. Novelli and J.-Y. Thibon, <a href="http://arXiv.org/abs/math.CO/0401089">The Algebra of Binary Search Trees</a>, Theoretical Computer Science, 339 (2005), 129-165.

%H T. Hyde and J. Lagarias <a href="http://arxiv.org/abs/1604.05359">Polynomial splitting measures and cohomology of the pure braid group</a>, arXiv preprint arXiv:1604.05359 [math.RT], 2016.

%H MathOverflow, <a href="https://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra/150935#150935">Motivation of Virasoro algebra</a>, an answer by Tom Copeland to an MO question posed in 2012.

%H Romeo Mestrovic, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014.

%H Robert E. Moritz, <a href="/A001701/a001701.pdf">On the sum of products of n consecutive integers</a>, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

%H M. D. Schmidt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">Generalized j-Factorial Functions, Polynomials, and Applications </a>, J. Int. Seq. 13 (2010), 10.6.7.

%H M. Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Spivey/spivey31.html">On Solutions to a General Combinatorial Recurrence</a>, J. Int. Seq. 14 (2011) # 11.9.7.

%H K. Yeats, <a href="https://www.math.uwaterloo.ca/~kayeats/teaching/co739/A%2BCombinatorial%2BPerspective%2Bon%2BQuantum%2BF.pdf">A Combinatorial Perspective on Quantum Field Theory</a>, SpringerBriefs in Mathematical Physics, Vol. 15, 2017.

%F With P(n,t) = Sum_{k=0..n-1} T(n,k+1) * t^k = 1*(1+t)*(1+2t)...(1+(n-1)*t) and P(0,t)=1, exp[P(.,t)*x] = (1-tx)^(-1/t). T(n,k+1) = (1/k!) (D_t)^k (D_x)^n [ (1-tx)^(-1/t) - 1 ] evaluated at t=x=0. (1-tx)^(-1/t) - 1 is the e.g.f. for a plane m-ary tree when t=m-1. See Bergeron et al. in "Varieties of Increasing Trees". - _Tom Copeland_, Dec 09 2007

%F First comment and formula above rephrased as o.g.f. for row n: Product_{i=0...n} (1+i*x). - _Geoffrey Critzer_, Feb 04 2011

%F n-th row polynomials with alternate signs are the characteristic polynomials of the (n-1)x(n-1) matrices with 1's in the superdiagonal, (1,2,3,...) in the main diagonal, and the rest zeros. For example, the characteristic polynomial of [1,1,0; 0,2,1; 0,0,3] is x^3 - 6*x^2 + 11*x - 6. - _Gary W. Adamson_, Jun 28 2011

%F E.g.f.: A(x,y) = x*y/(1 - x*y)^(1 + 1/y) = Sum_{n>=1, k=1..n} T(n,k)*x^n*y^k/(n-1)!. - _Paul D. Hanna_, Jul 21 2011

%F With F(x,t) = (1-t*x)^(-1/t) - 1 an e.g.f. for the row polynomials P(n,t) of A094638 with P(0,t)=0, G(x,t)= [1-(1+x)^(-t)]/t is the comp. inverse in x. Consequently, with H(x,t) = 1/(dG(x,t)/dx) = (1+x)^(t+1),

%F P(n,t) = [(H(x,t)*d/dx)^n] x evaluated at x=0; i.e.,

%F F(x,t) = exp[x*P(.,t)] = exp[x*H(u,t)*d/du] u, evaluated at u = 0.

%F Also, dF(x,t)/dx = H(F(x,t),t). - _Tom Copeland_, Sep 20 2011

%F T(n,k) = |A008276(n,k)|. - _R. J. Mathar_, May 19 2016

%F The row polynomials of this entry are the reversed row polynomials of A143491 multiplied by (1+x). E.g., (1+x)(1 + 5x + 6x^2) = (1 + 6x + 11x^2 + 6x^3). - _Tom Copeland_, Dec 11 2016

%F Regarding the row e.g.f.s in Copeland's 2007 formulas, e.g.f.s for A001710, A001715, and A001720 give the compositional inverses of the e.g.f. here for t = 2, 3, and 4 respectively. - _Tom Copeland_, Dec 28 2019

%e Triangle starts:

%e 1;

%e 1, 1;

%e 1, 3, 2;

%e 1, 6, 11, 6;

%e 1, 10, 35, 50, 24;

%e ...

%p T:=(n,k)->abs(Stirling1(n,n+1-k)): for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form. # _Emeric Deutsch_, Aug 14 2006

%t Table[CoefficientList[Series[Product[1 + i x, {i,n}], {x,0,20}], x], {n,0,6}] (* _Geoffrey Critzer_, Feb 04 2011 *)

%t Table[Abs@StirlingS1[n, n-k+1], {n, 10}, {k, n}]//Flatten (* _Michael De Vlieger_, Aug 29 2015 *)

%o (PARI) {T(n,k)=if(n<1 || k>n,0,(n-1)!*polcoeff(polcoeff(x*y/(1 - x*y+x*O(x^n))^(1 + 1/y),n,x),k,y))} /* _Paul D. Hanna_, Jul 21 2011 */

%o (Maxima) create_list(abs(stirling1(n+1,n-k+1)),n,0,10,k,0,n); / * _Emanuele Munarini_, Jun 01 2012 */

%o (Haskell)

%o a094638 n k = a094638_tabl !! (n-1) !! (k-1)

%o a094638_row n = a094638_tabl !! (n-1)

%o a094638_tabl = map reverse a130534_tabl

%o -- _Reinhard Zumkeller_, Aug 01 2014

%o (Magma) [(-1)^(k+1)*StirlingFirst(n,n-k+1): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Dec 29 2019

%o (Sage) [[stirling_number1(n, n-k+1) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, Dec 29 2019

%o (GAP) Flat(List([1..10], n-> List([1..n], k-> Stirling1(n,n-k+1) ))); # _G. C. Greubel_, Dec 29 2019

%Y A008276 gives the (signed) Stirling numbers of the first kind.

%Y Cf. A000108, A014137, A001246, A033536, A000984, A094639, A006134, A082894, A002897, A079727, A000217 (2nd column), A000914 (3rd column), A001303 (4th column), A000915 (5th column), A053567 (6th column), A000142 (row sums).

%Y Triangle sums (see the comments): A124380 (Kn11), A001710 (Fi1, Fi2). - _Johannes W. Meijer_, Apr 20 2011

%Y Cf. A121586, A130534, A143491.

%K easy,nonn,tabl

%O 1,5

%A _André F. Labossière_, May 17 2004

%E Edited by _Emeric Deutsch_, Aug 14 2006