%I M2739 N1100 #423 Oct 02 2024 04:25:34

%S 1,3,8,20,48,112,256,576,1280,2816,6144,13312,28672,61440,131072,

%T 278528,589824,1245184,2621440,5505024,11534336,24117248,50331648,

%U 104857600,218103808,452984832,939524096,1946157056,4026531840,8321499136,17179869184,35433480192

%N a(n) = (n+2)*2^(n-1).

%C Number of parts in all compositions (ordered partitions) of n + 1. For example, a(2) = 8 because in 3 = 2 + 1 = 1 + 2 = 1 + 1 + 1 we have 8 parts. Also number of compositions (ordered partitions) of 2n + 1 with exactly 1 odd part. For example, a(2) = 8 because the only compositions of 5 with exactly 1 odd part are 5 = 1 + 4 = 2 + 3 = 3 + 2 = 4 + 1 = 1 + 2 + 2 = 2 + 1 + 2 = 2 + 2 + 1. - _Emeric Deutsch_, May 10 2001

%C Binomial transform of natural numbers [1, 2, 3, 4, ...].

%C For n >= 1 a(n) is also the determinant of the n X n matrix with 3's on the diagonal and 1's elsewhere. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001

%C The arithmetic mean of first n terms of the sequence is 2^(n-1). - _Amarnath Murthy_, Dec 25 2001, corrected by _M. F. Hasler_, Dec 17 2016

%C Also the number of "winning paths" of length n across an n X n Hex board. Satisfies the recursion a(n) = 2a(n-1) + 2^(n-2). - David Molnar (molnar(AT)stolaf.edu), Apr 10 2002

%C Diagonal in A053218. - _Benoit Cloitre_, May 08 2002

%C Let M_n be the n X n matrix m_(i, j) = 1 + abs(i-j) then det(M_n) = (-1)^(n-1)*a(n-1). - _Benoit Cloitre_, May 28 2002

%C Absolute value of determinant of n X n matrix of form: [1 2 3 4 5 / 2 1 2 3 4 / 3 2 1 2 3 / 4 3 2 1 2 / 5 4 3 2 1]. - _Benoit Cloitre_, Jun 20 2002

%C Number of ones in all (n+1)-bit integers (cf. A000120). - _Ralf Stephan_, Aug 02 2003

%C This sequence also emerges as a floretion force transform of powers of 2 (see program code). Define a(-1) = 0 (as the sequence is returned by FAMP). Then a(n-1) + A098156(n+1) = 2*a(n) (conjecture). - _Creighton Dement_, Mar 14 2005

%C This sequence gives the absolute value of the determinant of the Toeplitz matrix with first row containing the first n integers. - _Paul Max Payton_, May 23 2006

%C Equals sums of rows right of left edge of A102363 divided by three, + 2^K. - David G. Williams (davidwilliams(AT)paxway.com), Oct 08 2007

%C If X_1, X_2, ..., X_n are 2-blocks of a (2n+1)-set X then, for n >= 1, a(n) is the number of (n+1)-subsets of X intersecting each X_i, (i = 1, 2, ..., n). - _Milan Janjic_, Nov 18 2007

%C Also, a(n-1) is the determinant of the n X n matrix with A[i, j] = n - |i-j|. - _M. F. Hasler_, Dec 17 2008

%C 1/2 the number of permutations of 1 .. (n+2) arranged in a circle with exactly one local maximum. - _R. H. Hardin_, Apr 19 2009

%C The first corrector line for transforming 2^n offset 0 with a leading 1 into the Fibonacci sequence. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009

%C a(n) is the number of runs of consecutive 1's in all binary sequences of length (n+1). - _Geoffrey Critzer_, Jul 02 2009

%C Let X_j (0 < j <= 2^n) all the subsets of N_n; m(i, j) := if {i} in X_j then 1 else 0. Let A = transpose(M).M; Then a(i, j) = (number of elements)|X_i intersect X_j|. Determinant(X*I-A) = (X-(n+1)*2^(n-2))*(X-2^(n-2))^(n-1)*X^(2^n-n).

%C Eigenvector for (n+1)*2^(n-2) is V_i=|X_i|.

%C Sum_{k=1..2^n} |X_i intersect X_k|*|X_k| = (n+1)*2^(n-2)*|X_i|.

%C Eigenvectors for 2^(n-2) are {line(M)[i] - line(M)[j], 1 <= i, j <= n}. - CLARISSE Philippe (clarissephilippe(AT)yahoo.fr), Mar 24 2010

%C The sequence b(n) = 2*A001792(n), for n >= 1 with b(0) = 1, is an elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 187, 190, 250 and 442, lead to the b(n) sequence. For the corner squares these vectors lead to the companion sequence A134401. - _Johannes W. Meijer_, Aug 15 2010

%C Equals partial sums of A045623: (1, 2, 5, 12, 28, ...); where A045623 = the convolution square of (1, 1, 2, 4, 8, 16, 32, ...). - _Gary W. Adamson_, Oct 26 2010

%C Equals (1, 2, 4, 8, 16, ...) convolved with (1, 1, 2, 4, 8, 16, ...); e.g., a(3) = 20 = (1, 1, 2, 4) dot (8, 4, 2, 1) = (8 + 4 + 4 + 4). - _Gary W. Adamson_, Oct 26 2010

%C This sequence seems to give the first x+1 nonzero terms in the sequence derived by subtracting the m-th term in the x_binacci sequence (where the first term is one and the y-th term is the sum of x terms immediately preceding it) from 2^(m-2). - _Dylan Hamilton_, Nov 03 2010

%C Recursive formulas for a(n) are in many cases derivable from its property wherein delta^k(a(n)) - a(n) = k*2^n where delta^k(a(n)) represents the k-th forward difference of a(n). Provable with a difference table and a little induction. - _Ethan Beihl_, May 02 2011

%C Let f(n,k) be the sum of numbers in the subsets of size k of {1, 2, ..., n}. Then a(n-1) is the average of the numbers f(n, 0), ... f(n, n). Example: (f(3, 1), f(3, 2), f(3, 3)) = (6, 12, 6), with average (6+12+6)/3. - _Clark Kimberling_, Feb 24 2012

%C a(n) is the number of length-2n binary sequences that contain a subsequence of ones with length n or more. To derive this result, note that there are 2^n sequences where the initial one of the subsequence occurs at entry one. If the initial one of the subsequence occurs at entry 2, 3, ..., or n + 1, there are 2^(n-1) sequences since a zero must precede the initial one. Hence a(n) = 2^n + n*2^(n-1)=(n+2)2^(n-1). An example is given in the example section below. - _Dennis P. Walsh_, Oct 25 2012

%C As the total number of parts in all compositions of n+1 (see the first line in Comments) the equivalent sequence for partitions is A006128. On the other hand, as the first differences of A001787 (see the first line in Crossrefs) the equivalent sequence for partitions is A138879. - _Omar E. Pol_, Aug 28 2013

%C a(n) is the number of spanning trees of the complete tripartite graph K_{n,1,1}. - _James Mahoney_, Oct 24 2013

%C a(n-1) = denominator of the mean (2n/(n+1), after reduction), of the compositions of n; numerator is given by A022998(n). - _Clark Kimberling_, Mar 11 2014

%C From _Tom Copeland_, Nov 09 2014: (Start)

%C The shifted array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interpolating o.g.f. (1-sqrt(1-4x/(1+(1-t)x)))/2 and inverse x(1-x)/(1+(t-1)x(1-x)). See A091867 for more info on this family. Here the interpolation is t=-3 (mod signs in the results).

%C Let C(x) = (1 - sqrt(1-4x))/2, an o.g.f. for the Catalan numbers A000108, with inverse Cinv(x) = x*(1-x) and P(x,t) = x/(1+t*x) with inverse P(x,-t).

%C Shifted o.g.f: G(x) = x*(1-x)/(1 - 4x*(1-x)) = P[Cinv(x),-4].

%C Inverse o.g.f: Ginv(x) = [1 - sqrt(1 - 4*x/(1+4x))]/2 = C[P(x, 4)] (signed shifted A001700). Cf. A030528. (End)

%C For n > 0, element a(n) of the sequence is equal to the gradients of the (n-1)-th row of Pascal triangle multiplied with the square of the integers from n+1,...,1. I.e., row 3 of Pascal's triangle 1,3,3,1 has gradients 1,2,0,-2,-1, so a(4) = 1*(5^2) + 2*(4^2) + 0*(3^2) - 2*(2^2) - 1*(1^2) = 48. - _Jens Martin Carlsson_, May 18 2017

%C Number of self-avoiding paths connecting all the vertices of a convex (n+2)-gon. - _Ivaylo Kortezov_, Jan 19 2020

%C a(n-1) is the total number of elements of subsets of {1,2,..,n} that contain n. For example, for n = 3, a(2) = 8, and the subsets of {1,2,3} that contain 3 are {3}, {1,3}, {2,3}, {1,2,3}, with a total of 8 elements. - _Enrique Navarrete_, Aug 01 2020

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D A. M. Stepin and A. T. Tagi-Zade, Words with restrictions, pp. 67-74 of Kvant Selecta: Combinatorics I, Amer. Math. Soc., 2001 (G_n on p. 70).

%H T. D. Noe, <a href="/A001792/b001792.txt">Table of n, a(n) for n = 0..500</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Marco Abrate, Stefano Barbero, Umberto Cerruti and Nadir Murru, <a href="http://dx.doi.org/10.1016/j.disc.2014.06.026">Colored compositions, Invert operator and elegant compositions with the "black tie"</a>, Discrete Math., Vol. 335 (2014), pp. 1-7. MR3248794.

%H Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="http://arxiv.org/abs/1409.6454">Colored compositions, Invert operator and elegant compositions with the "black tie"</a>, arXiv:1409.6454 [math.NT], 2014.

%H Milica Andelic, C. M. da Fonseca and A. Pereira, <a href="https://arxiv.org/abs/1609.04208">The mu-permanent, a new graph labeling, and a known integer sequence</a>, arXiv preprint arXiv:1609.04208 [math.CO], 2016.

%H Félix Balado and Guénolé C. M. Silvestre, <a href="https://arxiv.org/abs/2302.11532">Runs of Ones in Binary Strings</a>, arXiv:2302.11532 [math.CO], 2023. See pp. 6-7.

%H Neil J. Calkin, <a href="http://dx.doi.org/10.1016/0012-365X(94)90394-8">A curious binomial identity</a>, Discr. Math., Vol. 131, No. 1-3 (1994), pp. 335-337.

%H Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs., Vol. 3 (2000), #00.1.5.

%H Filippo Disanto and Simone Rinaldi, <a href="http://puma.dimai.unifi.it/22_1/2-disanto_rinaldi.pdf">Symmetric convex permutominoes and involutions</a>, PU. M. A., Vol. 22, No. 1 (2011), pp. 39-60.

%H Frank Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>.

%H David Eppstein, <a href="https://arxiv.org/abs/2303.00147">Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time</a>, arXiv:2303.00147 [cs.CG], 2023, pp. 2, 19.

%H Guillermo Esteban, Clemens Huemer and Rodrigo I. Silveira, <a href="https://arxiv.org/abs/2003.00524">New production matrices for geometric graphs</a>, arXiv:2003.00524 [math.CO], 2020.

%H M. Hirschhorn, <a href="http://dx.doi.org/10.1016/0012-365X(95)00086-C">Calkin's binomial identity</a>, Discr. Math., Vol. 159, No. 1-3 (1996), pp. 273-278.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=146">Encyclopedia of Combinatorial Structures 146</a>.

%H Milan Janjić, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>

%H Milan Janjić and Boris Petković, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013.

%H Milan Janjić and Boris Petković, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5

%H C. W. Jones, J. C. P. Miller, J. F. C. Conn and R. C. Pankhurst, <a href="http://dx.doi.org/10.1017/S0080454100006579">Tables of Chebyshev polynomials</a>, Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.

%H Sergey Kitaev and Jeffrey B. Remmel, <a href="https://pure.strath.ac.uk/portal/files/46917816/Kitaev_Remmel_JC2016_a_note_on_p_ascent_sequences.pdf">A note on p-Ascent Sequences</a>, Preprint, 2016.

%H Sergey Kitaev and Jeffrey Remmel, <a href="http://arxiv.org/abs/1503.00914">p-Ascent Sequences</a>, arXiv preprint arXiv:1503.00914 [math.CO], 2015.

%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%H Maohua Le, <a href="http://fs.gallup.unm.edu/ScientiaMagna2no1.pdf">Two Classes of Smarandache Determinants</a>, Scientia Magna, Vol. 2, No. 1 (2006), pp. 20-25.

%H Donatella Merlini and Massimo Nocentini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Merlini/merlini5.html">Algebraic Generating Functions for Languages Avoiding Riordan Patterns</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.

%H National Math Contest "Atanas Radev", <a href="http://www.mgyambol.com/docs/Broshura_ZMS_2020_v2.pdf">in problem 8.4 ("Задача 8.4" in Bulgarian), Jan 2020.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H Silvana Ramaj, <a href="https://digitalcommons.georgiasouthern.edu/etd/2273">New Results on Cyclic Compositions and Multicompositions</a>, Master's Thesis, Georgia Southern Univ., 2021.

%H John Riordan and N. J. A. Sloane, <a href="/A003471/a003471_1.pdf">Correspondence, 1974</a>.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%H I. Tasoulas, K. Manes, and A. Sapounakis, <a href="https://doi.org/10.37236/12144">Hamiltonian intervals in the lattice of binary paths</a>, Elect. J. Comb. (2024) Vol. 31, Issue 1, P1.39.

%H Jun Wang and Zhizheng Zhang, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00206-1">On extensions of Calkin's binomial identities</a>, Discrete Math., Vol. 274 (2004), 331-342.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F a(n) = (n+2)*2^(n-1).

%F G.f.: (1 - x)/(1 - 2*x)^2 = 2F1(1,3;2;2x).

%F a(n) = 4*a(n-1) - 4*a(n-2).

%F G.f. (-1 + (1-2*x)^(-2))/(x*2^2). - _Wolfdieter Lang_

%F a(n) = A018804(2^n). - _Matthew Vandermast_, Mar 01 2003

%F a(n) = Sum_{k=0..n+2} binomial(n+2, 2k)*k. - _Paul Barry_, Mar 06 2003

%F a(n) = (1/4)*A001787(n+2). - _Emeric Deutsch_, May 24 2003

%F With a leading 0, this is ((n+1)2^n - 0^n)/4 = Sum_{m=0..n} binomial(n - 1, m - 1)*m, the binomial transform of A004526(n+1). - _Paul Barry_, Jun 05 2003

%F a(n) = Sum_{k=0..n} binomial(n, k)*(k + 1). - _Lekraj Beedassy_, Jun 24 2004

%F a(n) = A000244(n) - A066810(n). - _Ross La Haye_, Apr 29 2006

%F Row sums of triangle A130585. - _Gary W. Adamson_, Jun 06 2007

%F Equals A125092 * [1/1, 1/2, 1/3, ...]. - _Gary W. Adamson_, Nov 16 2007

%F a(n) = (n+1)*2^n - n*2^(n-1). Equals A128064 * A000079. - _Gary W. Adamson_, Dec 28 2007

%F G.f.: F(3, 1; 2; 2x). - _Paul Barry_, Sep 03 2008

%F a(n) = 1 + Sum_{k=1..n} (n - k + 4)2^(n - k - 1). This follows from the result that the number of parts equal to k in all compositions of n is (n - k + 3)2^(n - k - 2) for 0 < k < n. - _Geoffrey Critzer_, Sep 21 2008

%F a(n) = 2^(n-1) + 2 a(n-1) ; a(n-1) = det(n - |i - j|)_{i, j = 1..n}. - _M. F. Hasler_, Dec 17 2008

%F a(n) = 2*a(n-1) + 2^(n-1). - _Philippe Deléham_, Apr 19 2009

%F a(n) = A164910(2^n). - _Gary W. Adamson_, Aug 30 2009

%F a(n) = Sum_{i=1..2^n} gcd(i, 2^n) = A018804(2^n). So we have: 2^0 * phi(2^n) + ... + 2^n * phi(2^0) = (n + 2)*2^(n-1), where phi is the Euler totient function. - _Jeffrey R. Goodwin_, Nov 11 2011

%F a(n) = Sum_{j=0..n} Sum_{i=0..n} binomial(n, i + j). - _Yalcin Aktar_, Jan 17 2012

%F Eigensequence of an infinite lower triangular matrix with 2^n as the left border and the rest 1's. - _Gary W. Adamson_, Jan 30 2012

%F G.f.: 1 + 2*x*U(0) where U(k) = 1 + (k + 1)/(2 - 8*x/(4*x + (k + 1)/U(k + 1))); (continued fraction, 3 - step). - _Sergei N. Gladkovskii_, Oct 19 2012

%F a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n,j). - _Peter Luschny_, Dec 03 2013

%F a(n) = Hyper2F1([-n, 2], [1], -1). - _Peter Luschny_, Aug 02 2014

%F G.f.: 1 / (1 - 3*x / (1 + x / (3 - 4*x))). - _Michael Somos_, Aug 26 2015

%F a(n) = -A053120(2+n, n), n >= 0, the negative of the third (sub)diagonal of the triangle of Chebyshev's T polynomials. - _Wolfdieter Lang_, Nov 26 2019

%F From _Amiram Eldar_, Jan 12 2021: (Start)

%F Sum_{n>=0} 1/a(n) = 8*log(2) - 4.

%F Sum_{n>=0} (-1)^n/a(n) = 4 - 8*log(3/2). (End)

%F E.g.f.: exp(2*x)*(1 + x). - _Stefano Spezia_, Jun 11 2021

%e a(0) = 1, a(1) = 2*1 + 1 = 3, a(2) = 2*3 + 2 = 8, a(3) = 2*8 + 4 = 20, a(4) = 2*20 + 8 = 48, a(5) = 2*48 + 16 = 112, a(6) = 2*112 + 32 = 256, ... - _Philippe Deléham_, Apr 19 2009

%e a(2) = 8 since there are 8 length-4 binary sequences with a subsequence of ones of length 2 or more, namely, 1111, 1110, 1101, 1011, 0111, 1100, 0110, and 0011. - _Dennis P. Walsh_, Oct 25 2012

%e G.f. = 1 + 3*x + 8*x^2 + 20*x^3 + 48*x^4 + 112*x^5 + 256*x^6 + 576*x^7 + ...

%p A001792 := n-> (n+2)*2^(n-1);

%p spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, B, B)}, labeled]: seq(combstruct[count](spec, size=n)/4, n=2..30); # _Zerinvary Lajos_, Oct 09 2006

%p A001792:=-(-3+4*z)/(2*z-1)^2; # _Simon Plouffe_ in his 1992 dissertation, which gives the sequence without the initial 1

%p G(x):=1/exp(2*x)*(1-x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(abs(f[n]),n=0..28 ); # _Zerinvary Lajos_, Apr 17 2009

%p a := n -> hypergeom([-n, 2], [1], -1);

%p seq(round(evalf(a(n),32)), n=0..31); # _Peter Luschny_, Aug 02 2014

%t matrix[n_Integer /; n >= 1] := Table[Abs[p - q] + 1, {q, n}, {p, n}]; a[n_Integer /; n >= 1] := Abs[Det[matrix[n]]] (* Josh Locker (joshlocker(AT)macfora.com), Apr 29 2004 *)

%t g[n_,m_,r_] := Binomial[n - 1, r - 1] Binomial[m + 1, r] r; Table[1 + Sum[g[n, k - n, r], {r, 1, k}, {n, 1, k - 1}], {k, 1, 29}] (* _Geoffrey Critzer_, Jul 02 2009 *)

%t a[n_] := (n + 2)*2^(n - 1); a[Range[0, 40]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 09 2011 *)

%t LinearRecurrence[{4, -4}, {1, 3}, 40] (* _Harvey P. Dale_, Aug 29 2011 *)

%t CoefficientList[Series[(1 - x) / (1 - 2 x)^2, {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 10 2014 *)

%t b[i_]:=i; a[n_]:=Abs[Det[ToeplitzMatrix[Array[b, n], Array[b, n]]]]; Array[a, 40] (* _Stefano Spezia_, Sep 25 2018 *)

%t a[n_]:=Hypergeometric2F1[2,-n+1,1,-1];Array[a,32] (* _Giorgos Kalogeropoulos_, Jan 04 2022 *)

%o (Octave, MATLAB) abs(det(toeplitz(1:n))) % _Paul Max Payton_, May 23 2006

%o (PARI) A001792(n)=(n+2)<<(n-1) \\ _M. F. Hasler_, Dec 17 2008

%o (Haskell)

%o a001792 n = a001792_list !! n

%o a001792_list = scanl1 (+) a045623_list

%o -- _Reinhard Zumkeller_, Jul 21 2013

%o (Magma) [(n+2)*2^(n-1): n in [0..40]]; // _Vincenzo Librandi_, Nov 10 2014

%o (GAP) List([0..35],n->(n+2)*2^(n-1)); # _Muniru A Asiru_, Sep 25 2018

%o (Python) for n in range(0,40): print(int((n+2)*2**(n-1)), end=' ') # _Stefano Spezia_, Oct 16 2018

%Y First differences of A001787.

%Y a(n) = A049600(n, 1), a(n) = A030523(n + 1, 1).

%Y Cf. A053113.

%Y Row sums of triangles A008949 and A055248.

%Y a(n) = -A039991(n+2, 2).

%Y Cf. A130584, A125092, A128064, A000079, A164910, A045623, A000120, A053120.

%Y Cf. A000108, A005043, A091867, A001700, A030528.

%Y If the exponent E in a(n) = Sum_{m=0..n} (Sum_{k=0..m} C(n,k))^E is 1, 2, 3, 4, 5 we get A001792, A003583, A007403, A294435, A294436 respectively.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_