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

0,3

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

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

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

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 *)

Crossrefs

Cf. A120844, A255802, A255835.

Cf. A005408.

Sequence in context: A105314 A200155 A002835 * A032288 A336975 A076859

Adjacent sequences: A253286 A253287 A253288 * A253290 A253291 A253292

Keyword

nonn

Author

Vaclav Kotesovec, Mar 07 2015

Status

approved