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Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)^2/4).
1

%I #6 Nov 09 2017 03:48:25

%S 1,1,10,46,191,740,2912,10941,40345,144703,509693,1761738,5993434,

%T 20076668,66329914,216307961,696990583,2220665661,7000973556,

%U 21853019072,67575353580,207111103623,629440843762,1897670845715,5677604053474,16863081962184,49736388996376,145714874857754

%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)^2/4).

%C Euler transform of A000537.

%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>

%F G.f.: Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)^2/4).

%F a(n) ~ exp(-Zeta(3) / (16*Pi^2) + 741*Zeta(5) / (1600*Pi^4) - 250047*Zeta(5)^3 / (5*Pi^14) + 10207918728 * Zeta(5)^5 / (5*Pi^24) + Zeta'(-3)/2 + (-7*(7/2)^(1/6) * Pi / (3200 * 3^(2/3)) + 9261 * 3^(1/3) * (7/2)^(1/6) * Zeta(5)^2 / (40*Pi^9) - 22754277 * 3^(1/3) * (7/2)^(1/6) * Zeta(5)^4 / (2*Pi^19)) * n^(1/6) + (-21 * 3^(2/3) * (7/2)^(1/3) * Zeta(5) / (20*Pi^4) + 31752 * 6^(2/3) * 7^(1/3) * Zeta(5)^3/Pi^14) * n^(1/3) + (sqrt(7/2)*Pi/60 - 567*sqrt(14)*Zeta(5)^2 / Pi^9) * sqrt(n) + 9 * 3^(1/3) * (7/2)^(2/3) * Zeta(5) / Pi^4 * n^(2/3) + 2 * (2/7)^(1/6) * 3^(2/3) * Pi/5 * n^(5/6)) / (2^(1321/1440) * 3^(479/720) * 7^(119/1440) * n^(839/1440) * Pi^(1/240)). - _Vaclav Kotesovec_, Nov 09 2017

%t nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k^2 (k + 1)^2/4), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000294, A000537, A023872, A279215.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 19 2017