%I #11 Jan 16 2025 19:37:53
%S 1,5,22,79,259,777,2201,5911,15239,37865,91224,213741,488759,1093173,
%T 2396934,5160756,10928181,22787949,46848176,95046026,190466354,
%U 377295743,739319876,1433974869,2754597217,5243308562,9894376295,18517966608,34386781020,63378252332
%N G.f.: Product_{k>=1} 1/(1-x^k)^(2*k+3).
%H Vaclav Kotesovec, <a href="/A255802/b255802.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ Zeta(3)^(13/18) * exp(1/6 - Pi^4/(96*Zeta(3)) + Pi^2 * n^(1/3) / (2^(5/3) * Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)) / (A^2 * 2^(5/9) * 3^(1/2) * Pi^2 * n^(11/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .
%F G.f.: exp(Sum_{k >= 1} (2*sigma_2(k) + 3*sigma_1(k))*x^k/k) = 1 + 5*x + 22*x^2 + 29*x^3 + 777*x^4 + .... - _Peter Bala_, Jan 16 2025
%p with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> 2*n+3): seq(a(n), n=0..50); # after _Alois P. Heinz_
%p with(numtheory):
%p series(exp(add((2*sigma[2](k) + 3*sigma[1](k))*x^k/k, k = 1..30)), x, 31):
%p seq(coeftayl(%, x = 0, n), n = 0..30); # _Peter Bala_, Jan 16 2025
%t nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(2*k+3),{k,1,nmax}],{x,0,nmax}],x]
%Y Cf. A005380, A120844, A217093, A253289, A255271, A255803, A255834, A255835, A255836, A363601, A363602.
%K nonn,easy
%O 0,2
%A _Vaclav Kotesovec_, Mar 07 2015