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

%I A253289 #52 Jan 16 2025 11:30:07
%S A253289 1,1,4,9,22,46,103,208,431,849,1671,3195,6079,11321,20937,38146,68931,
%T A253289 123121,218212,383019,667425,1153544,1980268,3375394,5717773,9624541,
%U A253289 16108496,26807662,44379189,73089219,119789926,195401275,317309532,513025167,826000651
%N A253289 G.f.: Product_{k>=1} 1/(1-x^k)^(2*k-1).
%C A253289 a(n) is the number of partitions of n where there are 2*k-1 sorts of parts k. - _Joerg Arndt_, Aug 15 2020
%H A253289 Vaclav Kotesovec, <a href="/A253289/b253289.txt">Table of n, a(n) for n = 0..1000</a>
%H A253289 Vaclav Kotesovec, <a href="/A253289/a253289.jpg">Graph - The asymptotic ratio</a>
%F A253289 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... .
%F A253289 G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^2)). - _Ilya Gutkovskiy_, Jun 07 2018
%F A253289 Euler transform of A005408 (the odd numbers). - _Georg Fischer_, Aug 15 2020
%F A253289 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
%p A253289 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_
%p A253289 with(numtheory):
%p A253289 series(exp(add((2*sigma[2](k) - sigma[1](k))*x^k/k, k = 1..30)), x, 31):
%p A253289 seq(coeftayl(%, x = 0, n), n = 0..30); # _Peter Bala_, Jan 16 2025
%t A253289 nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(2*k-1),{k,1,nmax}],{x,0,nmax}],x]
%t A253289 (* Using EulerTransforms from 'Transforms'. *)
%t A253289 Prepend[EulerTransform[Table[2k + 1, {k, 0, 20}]], 1] (* _Peter Luschny_, Aug 15 2020 *)
%Y A253289 Cf. A005380, A120844, A217093, A253289, A255271, A255802, A255803, A255835, A255836, A363601, A363602.
%K A253289 nonn,easy
%O A253289 0,3
%A A253289 _Vaclav Kotesovec_, Mar 07 2015