A352694 -id:A352694 - cp's OEIS frontend

A352693 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} sigma_2(k) * x^k/k!).

1, 1, 7, 46, 455, 5406, 78172, 1312116, 25214479, 544777183, 13080808752, 345471545728, 9953804592152, 310687941345796, 10443489230611052, 376122782541917166, 14449157656748079247, 589772212576633845886, 25488817336672959449725

Offset: 0

Seiichi Manyama, Mar 29 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..393

Cf. A340903, A340904.

Cf. A001157, A320649, A352694.

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

a(n) = if(n==0, 1, sum(k=1, n, sigma(k, 2)*binomial(n, k)*a(n-k)));

a(0) = 1; a(n) = Sum_{k=1..n} sigma_2(k) * binomial(n,k) * a(n-k).

A352843 Expansion of e.g.f. exp(Sum_{k>=1} sigma_k(k) * x^k/k!).

Original entry on oeis.org

1, 1, 6, 44, 491, 6597, 110652, 2144606, 47988524, 1206275925, 33777572464, 1040200674416, 34967153135940, 1273241146218823, 49928549099500206, 2097300313258417056, 93953420539864844743, 4470694981375022862697, 225184078001798318202935

Offset: 0

Seiichi Manyama, Apr 05 2022

nonn

Exponential transform of A023887.

Cf. A295739, A274804, A352694.

Cf. A023887, A352841, A352842, A202477.

nmax = 20; CoefficientList[Series[E^(Sum[DivisorSigma[k, k]*x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2022 *)

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

a(n) = if(n==0, 1, sum(k=1, n, sigma(k, k)*binomial(n-1, k-1)*a(n-k)));

a(0) = 1; a(n) = Sum_{k=1..n} sigma_k(k) * binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

A352693 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} sigma_2(k) * x^k/k!).

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A352843 Expansion of e.g.f. exp(Sum_{k>=1} sigma_k(k) * x^k/k!).

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