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A317017 Expansion of Product_{k>=1} 1/(1 - x^k)^((3*k+1)*binomial(k+2,3)/4).

%I A317017 #9 Apr 03 2019 03:00:09
%S A317017 1,1,8,33,126,441,1571,5338,17900,58359,187134,588966,1826537,5580784,
%T A317017 16831549,50135506,147650112,430187724,1240908651,3545808444,
%U A317017 10042128414,28201458999,78567720054,217225969695,596254164090,1625343030654,4401332943214,11843216471115,31674767502610
%N A317017 Expansion of Product_{k>=1} 1/(1 - x^k)^((3*k+1)*binomial(k+2,3)/4).
%C A317017 Euler transform of A001296.
%H A317017 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A317017 G.f.: Product_{k>=1} 1/(1 - x^k)^A001296(k).
%F A317017 G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^k)^5)).
%F A317017 a(n) ~ Pi^(1/288)/(2 * 3^(577/864) * 7^(145/1728) * n^(1009/1728)) * exp(1/144 - (1/12-Zeta'(-1))/12 - (11 * Zeta(3))/(80 * Pi^2) + (1383 * Zeta(5))/(640 * Pi^4) + (11025 * Zeta(3) * Zeta(5)^2)/(2 * Pi^12) - (694575 * Zeta(5)^3)/(2 * Pi^14) + (13127467500 * Zeta(5)^5)/Pi^24 + (5 * Zeta'(-3))/12 + ((-21 * 3^(1/3) * 7^(1/6) * Pi)/6400 - (35 * 3^(1/3) * 7^(1/6) * Zeta(3) * Zeta(5))/(2 * Pi^7) + (15435 * 3^(1/3) * 7^(1/6) * Zeta(5)^2)/(16 * Pi^9) - (175573125 * 3^(1/3) * 7^(1/6) * Zeta(5)^4)/(4 * Pi^19)) * n^(1/6) + (((7/3)^(1/3) * Zeta(3))/(4 * Pi^2) - (21 * 3^(2/3) * 7^(1/3) * Zeta(5))/(8 * Pi^4) + (147000 * 3^(2/3) * 7^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7) * Pi)/40 - (1575 * sqrt(7) * Zeta(5)^2)/Pi^9) * sqrt(n) + ((15 * 3^(1/3) * 7^(2/3) * Zeta(5))/(2 * Pi^4)) * n^(2/3) + ((2 * 3^(2/3) * Pi)/(5 * 7^(1/6))) * n^(5/6)). - _Vaclav Kotesovec_, Jul 28 2018
%p A317017 a:=series(mul(1/(1-x^k)^((3*k+1)*binomial(k+2,3)/4),k=1..100),x=0,29): seq(coeff(a,x,n),n=0..28); # _Paolo P. Lava_, Apr 02 2019
%t A317017 nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^((3 k + 1) Binomial[k + 2, 3]/4), {k, 1, nmax}], {x, 0, nmax}], x]
%t A317017 nmax = 28; CoefficientList[Series[Exp[Sum[x^k (1 + 2 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
%t A317017 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (3 d + 1)/24, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]
%Y A317017 Cf. A000391, A001296, A278768, A279216, A305653, A317019, A317020, A317021.
%K A317017 nonn
%O A317017 0,3
%A A317017 _Ilya Gutkovskiy_, Jul 19 2018