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

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

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

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 *)

Prog

(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)))

Crossrefs

Cf. A061255, A381680.

Cf. A000056, A084218, A156733.

Sequence in context: A199921 A192596 A383834 * A055899 A218956 A139876

Adjacent sequences: A381676 A381677 A381678 * A381680 A381681 A381682

Keyword

nonn

Author

Seiichi Manyama, Mar 04 2025

Status

approved