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%I #12 Feb 16 2025 08:33:37
%S 1,1,8,30,108,357,1205,3838,12083,36896,110828,326281,946086,2700026,
%T 7602642,21128513,58028309,157588912,423534324,1127102360,2971764946,
%U 7766890826,20131080168,51766513279,132117237595,334770353022,842462217948,2106183375971,5232414548275,12920429411759,31719180847831
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(4*k-1)/6).
%C Euler transform of the hexagonal pyramidal numbers (A002412).
%H Vaclav Kotesovec, <a href="/A279217/b279217.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexagonalPyramidalNumber.html">Hexagonal Pyramidal Number</a>
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(4*k-1)/6).
%F a(n) ~ exp(-Zeta'(-1)/6 - Zeta(3)/(8*Pi^2) - Pi^16/(199065600000*Zeta(5)^3) - Pi^8*Zeta(3)/(6912000*Zeta(5)^2) - Zeta(3)^2/(1440*Zeta(5)) + 2*Zeta'(-3)/3 + (Pi^12/(172800000*2^(4/5)*Zeta(5)^(11/5)) + Pi^4*Zeta(3)/(7200*2^(4/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(288000*2^(3/5)*Zeta(5)^(7/5)) - Zeta(3)/(12*2^(3/5)*Zeta(5)^(2/5))) * n^(2/5) + (Pi^4/(360*2^(2/5)*Zeta(5)^(3/5))) * n^(3/5) + 5*(Zeta(5)/2)^(1/5)/2 * n^(4/5)) * Zeta(5)^(173/1800) / (2^(26/225) * sqrt(5*Pi) * n^(1073/1800)). - _Vaclav Kotesovec_, Dec 08 2016
%t nmax=30; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1)(4 k - 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000335, A002412, A279215, A279216, A279218, A279219.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Dec 08 2016