A253289 G.f.: Product_{k>=1} 1/(1-x^k)^(2*k-1).

1, 1, 4, 9, 22, 46, 103, 208, 431, 849, 1671, 3195, 6079, 11321, 20937, 38146, 68931, 123121, 218212, 383019, 667425, 1153544, 1980268, 3375394, 5717773, 9624541, 16108496, 26807662, 44379189, 73089219, 119789926, 195401275, 317309532, 513025167, 826000651

Offset: 0

Author

Vaclav Kotesovec, Mar 07 2015

Keywords

• nonn
• easy

Comments

a(n) is the number of partitions of n where there are 2*k-1 sorts of parts k. - Joerg Arndt, Aug 15 2020

Links

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

Crossrefs

Cf. A005380, A120844, A217093, 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-> 2*n-1): seq(a(n), n=0..50); # after Alois P. Heinz
with(numtheory):
series(exp(add((2*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)^(2*k-1),{k,1,nmax}],{x,0,nmax}],x]
(* Using EulerTransforms from 'Transforms'. *)
Prepend[EulerTransform[Table[2k + 1, {k, 0, 20}]], 1] (* Peter Luschny, Aug 15 2020 *)

Formula

a(n) ~ 2^(1/9) * Zeta(3)^(1/18) * exp(1/6 - Pi^4/(864*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(5/3) * Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)) / (A^2 * 3^(1/2) * n^(5/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Jun 07 2018

Euler transform of A005408 (the odd numbers). - Georg Fischer, Aug 15 2020

G.f.: exp(Sum_{k >= 1} (2*sigma_2(k) - sigma_1(k))*x^k/k) = 1 + x + 4*x^2 + 9*x^3 + 22*x^4 + .... - Peter Bala, Jan 16 2025