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A279216 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).

%I A279216 #12 Feb 16 2025 08:33:37
%S A279216 1,1,7,25,86,269,862,2606,7812,22704,64989,182356,504414,1373694,
%T A279216 3693367,9804435,25733084,66808578,171719539,437183839,1103143657,
%U A279216 2760037810,6850400668,16873338215,41260373472,100196920196,241712863504,579416535973,1380517695672,3270075208145,7702580246941
%N A279216 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).
%C A279216 Euler transform of the pentagonal pyramidal numbers (A002411).
%H A279216 Vaclav Kotesovec, <a href="/A279216/b279216.txt">Table of n, a(n) for n = 0..1000</a>
%H A279216 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A279216 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A279216 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A279216 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalPyramidalNumber.html">Pentagonal Pyramidal Number</a>
%H A279216 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%F A279216 G.f.: Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).
%F A279216 a(n) ~ exp(-Zeta(3)/(8*Pi^2) - Pi^16/(83980800000*Zeta(5)^3) + Zeta'(-3)/2 + (Pi^12/(97200000*2^(2/5)*3^(1/5)*Zeta(5)^(11/5))) * n^(1/5) + (-Pi^8/(108000*2^(4/5)*3^(2/5)*Zeta(5)^(7/5))) * n^(2/5) + (Pi^4/(180*2^(1/5)*(3*Zeta(5))^(3/5))) * n^(3/5) + ((5*(3*Zeta(5))^(1/5))/(2^(8/5))) * n^(4/5)) * (3*Zeta(5))^(119/1200) / (2^(181/600) * sqrt(5*Pi) * n^(719/1200)). - _Vaclav Kotesovec_, Dec 08 2016
%t A279216 nmax=30; CoefficientList[Series[Product[1/(1 - x^k)^(k^2 (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A279216 Cf. A000335, A002411, A279215, A279217, A279218, A279219.
%K A279216 nonn
%O A279216 0,3
%A A279216 _Ilya Gutkovskiy_, Dec 08 2016