A255271 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k+1).

1, 4, 17, 58, 186, 546, 1532, 4082, 10502, 26096, 63075, 148536, 342096, 771744, 1709299, 3721792, 7978972, 16860328, 35155475, 72393580, 147351112, 296657196, 591141762, 1166570452, 2281101159, 4421781894, 8500806341, 16214549920, 30696683828

Offset: 0

Author

Vaclav Kotesovec, Mar 07 2015

Keywords

• nonn
• easy

Comments

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Graph - The asymptotic ratio

Crossrefs

Cf. A005380, A120844, A253289, A255271, A255802, A255803, A255835, A255836, A363601, A363602.

Programs

Maple

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
with(numtheory):
series(exp(add((3*sigma[2](k) + sigma[1](k))*x^k/k, k = 1..30)), x, 31):
seq(coeftayl(%, x = 0, n), n = 0..30); # Peter Bala, Jan 16 2025

Mathematica

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(3*k+1),{k,1,nmax}],{x,0,nmax}],x]

Formula

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... .

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