%I #10 Jun 09 2018 00:18:50
%S 1,1,7,27,98,323,1085,3471,10998,33874,102737,305849,897899,2597822,
%T 7423408,20957775,58524868,161741013,442705279,1200718351,3228796864,
%U 8611973548,22793714865,59887897679,156252738062,404964879419,1042884107691,2669317020743,6792321636929
%N Expansion of Product_{k>=1} 1/(1 - x^k)^((k+1)*binomial(k+2,3)/2).
%C Euler transform of A002415, shifted left one place.
%H Alois P. Heinz, <a href="/A305653/b305653.txt">Table of n, a(n) for n = 0..2000</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^A002415(k+1).
%F G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^5)).
%F a(n) ~ exp(Zeta'(-1)/6 - Zeta(3) / (4*Pi^2) + 149*Zeta(5) / (32*Pi^4) + 15876 * Zeta(3) * Zeta(5)^2 / Pi^12 - 666792 * Zeta(5)^3 / Pi^14 + 108884466432 * Zeta(5)^5 / (5*Pi^24) + Zeta'(-3)/3 + (-7*(7/2)^(1/6) * Pi / (384*sqrt(3)) - 21 * 2^(5/6) * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5) / Pi^7 + 3087 * sqrt(3) * (7/2)^(1/6) * Zeta(5)^2 / (2*Pi^9) - 30339036 * 2^(5/6) * sqrt(3) * 7^(1/6) * Zeta(5)^4 / Pi^19) * n^(1/6) + ((7/2)^(1/3) * Zeta(3) / (2*Pi^2) - 21 * (7/2)^(1/3) * Zeta(5) / (2*Pi^4) + 254016 * 2^(2/3) * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (sqrt(7/6) * Pi / 12 - 756 * sqrt(42) * Zeta(5)^2 / Pi^9) * sqrt(n) + (9 * 2^(1/3) * 7^(2/3) * Zeta(5) / Pi^4) * n^(2/3) + (2 * (2/7)^(1/6) * sqrt(3) * Pi) / 5 * n^(5/6)) * Pi^(1/90) / (2^(247/270) * 3^(34/45) * 7^(23/270) * n^(79/135)). - _Vaclav Kotesovec_, Jun 08 2018
%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d^2*
%p (d+2)*(d+1)^2/12, d=numtheory[divisors](j)), j=1..n)/n)
%p end:
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Jun 07 2018
%t nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^((k + 1) Binomial[k + 2, 3]/2), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 28; CoefficientList[Series[Exp[Sum[x^k (1 + x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1)^2 (d + 2)/12, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]
%Y Cf. A000391, A002415, A023871, A279215, A290792.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jun 07 2018