%I #50 Jan 16 2025 19:37:46
%S 1,4,17,58,186,546,1532,4082,10502,26096,63075,148536,342096,771744,
%T 1709299,3721792,7978972,16860328,35155475,72393580,147351112,
%U 296657196,591141762,1166570452,2281101159,4421781894,8500806341,16214549920,30696683828
%N G.f.: Product_{k>=1} 1/(1-x^k)^(3*k+1).
%C In general, if g.f. = Product_{k>=1} 1/(1-x^k)^(m*k+c), m > 0, then a(n) ~ (m*Zeta(3))^(m/36 + c/6 + 1/6) * exp(m/12 - c^2 * Pi^4 / (432*m*Zeta(3)) + c * Pi^2 * n^(1/3) / (3 * 2^(4/3) * (m*Zeta(3))^(1/3)) + 3 * (m*Zeta(3))^(1/3) * n^(2/3) / 2^(2/3)) / (A^m * 2^(c/3 + 1/3 - m/36) * 3^(1/2) * Pi^((c+1)/2) * n^(m/36 + c/6 + 2/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Mar 08 2015
%H Vaclav Kotesovec, <a href="/A255271/b255271.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="/A255271/a255271.jpg">Graph - The asymptotic ratio</a>
%F a(n) ~ Zeta(3)^(5/12) * exp(1/4 - Pi^4/(1296*Zeta(3)) + Pi^2 * n^(1/3) / (6^(4/3) * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A^3 * 2^(7/12) * 3^(1/12) * Pi * n^(11/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .
%F G.f.: exp(Sum_{k >= 1} (3*sigma_2(k) + sigma_1(k))*x^k/k) = 1 + 4*x + 17*x^2 + 58*x^3 + 186*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-> 3*n+1): seq(a(n), n=0..50); # after _Alois P. Heinz_
%p with(numtheory):
%p series(exp(add((3*sigma[2](k) + 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)^(3*k+1),{k,1,nmax}],{x,0,nmax}],x]
%Y Cf. A005380, A120844, A253289, A255271, A255802, A255803, A255835, A255836, A363601, A363602.
%K nonn,easy
%O 0,2
%A _Vaclav Kotesovec_, Mar 07 2015