cp's OEIS Frontend

A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

%I A000294 M3393 N1372 #125 Feb 16 2023 12:23:58
%S A000294 1,1,4,10,26,59,141,310,692,1483,3162,6583,13602,27613,55579,110445,
%T A000294 217554,424148,820294,1572647,2992892,5652954,10605608,19765082,
%U A000294 36609945,67405569,123412204,224728451,407119735,733878402,1316631730,2351322765,4180714647,7401898452,13051476707,22922301583,40105025130,69909106888,121427077241,210179991927,362583131144
%N A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).
%C A000294 Number of partitions of n if there are k(k+1)/2 kinds of k (k=1,2,...). E.g., a(3)=10 because we have six kinds of 3, three kinds of 2+1 because there are three kinds of 2 and 1+1+1+1. - _Emeric Deutsch_, Mar 23 2005
%C A000294 Euler transform of the triangular numbers 1,3,6,10,...
%C A000294 Equals A028377: [1, 1, 3, 9, 19, 46, 100, ...] convolved with the aerated version of A000294: [1, 0, 1, 0, 4, 0, 10, 0, 26, 0, 59, ...]. - _Gary W. Adamson_, Jun 13 2009
%C A000294 The formula for p3(n) in the article by S. Finch (page 2) is incomplete, terms with n^(1/2) and n^(1/4) are also needed. These terms are in the article by Mustonen and Rajesh (page 2) and agree with my results, but in both articles the multiplicative constant is marked only as C, resp. c3(m). The following is a closed form of this constant: Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8)) / (A^(1/2) * 2^(157/96) * 15^(13/96)) = A255939 = 0.213595160470..., where A = A074962 is the Glaisher-Kinkelin constant and Zeta(3) = A002117. - _Vaclav Kotesovec_, Mar 11 2015 [The new version of "Integer Partitions" by S. Finch contains the missing terms, see pages 2 and 5. - _Vaclav Kotesovec_, May 12 2015]
%C A000294 Number of solid partitions of corner-hook volume n (see arXiv:2009.00592 among links for definition). E.g., a(2) = 1 because there is only one solid partition [[[2]]] with cohook volume 2; a(3) = 4 because we have three solid partitions with two 1's (entry at (1,1,1) contributes 1, another entry at (2,1,1) or (1,2,1) or (1,1,2) contributes 2 to corner-hook volume) and one solid partition with single entry 3 (which contributes 3 to the corner-hook volume). Namely as 3D arrays [[[1],[1]]],[[[1]],[[1]]],[[[1]],[[1]]], [[[3]]]. - _Alimzhan Amanov_, Jul 13 2021
%D A000294 R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139.
%D A000294 V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314.
%D A000294 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000294 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000294 Alois P. Heinz, <a href="/A000294/b000294.txt">Table of n, a(n) for n = 0..1000</a>
%H A000294 Alimzhan Amanov and Damir Yeliussizov, <a href="https://arxiv.org/abs/2009.00592">MacMahon's statistics on higher-dimensional partitions</a>, arXiv:2009.00592 [math.CO], 2020. Mentions this sequence.
%H A000294 A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, <a href="http://boltzmann.wdfiles.com/local--files/refined-counting/ABMM.pdf">Some computations for m-dimensional partitions</a>, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
%H A000294 A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, <a href="/A000219/a000219.pdf">Some computations for m-dimensional partitions</a>, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
%H A000294 R. Chandra, <a href="/A000294/a000294_1.pdf">Tables of solid partitions</a>, Proceedings of the Indian National Science Academy, 26 (1960), 134-139. [Annotated scanned copy]
%H A000294 Nicolas Destainville and Suresh Govindarajan, <a href="http://arxiv.org/abs/1406.5605">Estimating the asymptotics of solid partitions</a>, J. Stat. Phys. 158 (2015) 950-967; arXiv:1406.5605 [cond-mat.stat-mech], 2014.
%H A000294 Steven Finch, <a href="/A000219/a000219_1.pdf">Integer Partitions</a>, September 22, 2004, page 2. [Cached copy, with permission of the author]
%H A000294 Vaclav Kotesovec, <a href="/A000294/a000294.jpg">Graph - The asymptotic ratio</a>
%H A000294 Ville Mustonen and R. Rajesh, <a href="http://arXiv.org/abs/cond-mat/0303607">Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer</a>, J. Phys. A 36 (2003), no. 24, 6651-6659; arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.
%H A000294 V. S. Nanda, <a href="/A000294/a000294.pdf">Tables of solid partitions</a>, Proceedings of the Indian National Science Academy, 19 (1953), 313-314. [Annotated scanned copy]
%H A000294 Damir Yeliussizov, <a href="https://arxiv.org/abs/2302.04799">Bounds on the number of higher-dimensional partitions</a>, arXiv:2302.04799 [math.CO], 2023.
%F A000294 a(n) = (1/(2*n))*Sum_{k=1..n} (sigma[2](k)+sigma[3](k))*a(n-k). - _Vladeta Jovovic_, Sep 17 2002
%F A000294 a(n) ~ Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(7/4)*Pi^5) + 15^(1/2) * Zeta(3) * n^(1/2) / (2^(1/2)*Pi^2) + 2^(7/4) * Pi * n^(3/4) / (3*15^(1/4))) / (A^(1/2) * 2^(157/96) * 15^(13/96) * n^(61/96)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - _Vaclav Kotesovec_, Mar 11 2015
%F A000294 G.f.: exp(Sum_{k>=1} (sigma_2(k) + sigma_3(k))*x^k/(2*k)). - _Ilya Gutkovskiy_, Aug 21 2018
%p A000294 with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> n*(n+1)/2): seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 08 2008
%t A000294 a[0] = 1; a[n_] := a[n] = 1/(2*n)*Sum[(DivisorSigma[2, k]+DivisorSigma[3, k])*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 05 2014, after _Vladeta Jovovic_ *)
%t A000294 nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)/2),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Mar 11 2015 *)
%o A000294 (PARI) a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^3/k, x*O(x^n))), n)) \\ _Joerg Arndt_, Apr 16 2010
%o A000294 (SageMath) # uses[EulerTransform from A166861]
%o A000294 b = EulerTransform(lambda n: binomial(n+1, 2))
%o A000294 print([b(n) for n in range(37)]) # _Peter Luschny_, Nov 11 2020
%Y A000294 Cf. A000293, A007294, A007326, A255939, A028377.
%Y A000294 Cf. also A000041, A000219, A000335, A000391, A000417, A000428, A255965.
%Y A000294 Cf. also A278403 (log of o.g.f.).
%K A000294 nonn,easy
%O A000294 0,3
%A A000294 _N. J. A. Sloane_
%E A000294 More terms from _Sascha Kurz_, Aug 15 2002