A255803 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k+2).

1, 5, 23, 86, 295, 926, 2748, 7732, 20891, 54401, 137355, 337249, 808043, 1893402, 4348634, 9805669, 21741925, 47463473, 102133056, 216841459, 454648373, 942113618, 1930779697, 3915946921, 7864385266, 15647363323, 30858285440, 60345383394, 117065924679

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

Crossrefs

Cf. A000219 (k), A005380 (k+1), A052847 (k-1), A120844 (2k+1), A253289 (2k-1), A255802 (2k+3), A255271 (3k+1).

Cf. A255834 - A255837, 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+2): seq(a(n), n=0..50); # after Alois P. Heinz
with(numtheory):
series(exp(add((3*sigma[2](k) + 2*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+2),{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ Zeta(3)^(7/12) * 3^(1/12) * exp(1/4 - Pi^4/(324*Zeta(3)) + Pi^2 * n^(1/3) / (3^(4/3) * (2*Zeta(3))^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A^3 * 2^(11/12) * Pi^(3/2) * n^(13/12)), 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) + 2*sigma_1(k))*x^k/k) = 1 + 5*x + 23*x^2 + 86*x^3 + 295*x^4 + .... - Peter Bala, Jan 16 2025