A381679 Euler transform of A000056.
1, 1, 7, 31, 100, 364, 1152, 3864, 12102, 37358, 113618, 337562, 990798, 2857926, 8144334, 22902470, 63660695, 175026047, 476242001, 1283435153, 3427047146, 9072455146, 23820491998, 62057045134, 160471504373, 412022656517, 1050740365571, 2662223436203
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Programs
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Mathematica
a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[4, k^2]/DivisorSigma[2, k^2]*a[n-k], {k, 1, n}]/n; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Mar 04 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 4)/sigma(k^2, 2)*x^k/k)))
Formula
G.f.: 1/Product_{k>=1} (1 - x^k)^A000056(k).
G.f.: exp( Sum_{k>=1} sigma_4(k^2)/sigma_2(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_4(k^2)/sigma_2(k^2) * a(n-k).
a(n) ~ exp(5*(3*zeta(5)/zeta(3))^(1/5) * n^(4/5) / 2^(7/5) - 1/10 - 12*zeta'(-3)) * A^(6/5) * (3*zeta(5)/zeta(3))^(3/25) / (2^(7/50) * sqrt(5*Pi) * n^(31/50)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 04 2025