A381679 -id:A381679 - cp's OEIS frontend

A381680 Euler transform of A115224.

1, 1, 29, 263, 1565, 11217, 74412, 482638, 2987123, 18066149, 107415185, 623612637, 3552605428, 19882256022, 109518424910, 594290145192, 3179607733480, 16790129919934, 87573088547032, 451477766533886, 2302069862201553, 11616226357007259, 58036597014533469

Offset: 0

Seiichi Manyama, Mar 04 2025

nonn

Cf. A061255, A381679.

Cf. A115224, A084220.

a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[6, k^2]/DivisorSigma[3, k^2]*a[n-k], {k, 1, n}]/n; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Mar 04 2025 *)

my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 6)/sigma(k^2, 3)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^A115224(k).
G.f.: exp( Sum_{k>=1} sigma_6(k^2)/sigma_3(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_6(k^2)/sigma_3(k^2) * a(n-k).
log(a(n)) ~ 7 * 5^(2/7) * zeta(7)^(1/7) * n^(6/7) / (2^(2/7) * 3^(3/7) * Pi^(4/7)). - Vaclav Kotesovec, Mar 04 2025

A381709 Euler transform of n^3 * A065960(n).

Original entry on oeis.org

1, 1, 137, 2351, 29075, 408429, 5957562, 76590384, 955079422, 11831378688, 142650905559, 1668991927795, 19144774189917, 215790313316371, 2388025355854986, 25973791505651972, 278176027053878678, 2936495245822593766, 30573379794788083289, 314185573464039742503

Offset: 0

Seiichi Manyama, Mar 04 2025

nonn

Cf. A156303, A156733, A381170.

Cf. A023872, A065960, A381679.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 4)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^(k^3 * A065960(k)).
G.f.: exp( Sum_{k>=1} sigma_4(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_4(k^2) * a(n-k).

cp's OEIS Frontend

A381680 Euler transform of A115224.

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