A354341 -id:A354341 - cp's OEIS frontend

A354338 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} 1/(d * (k/d)!) )/(n-k)!.

1, 4, 12, 41, 145, 742, 3962, 27659, 215131, 1996356, 17300360, 218809109, 2421142269, 31105286682, 427776526574, 6964677268087, 97708052695959, 1856379196278120, 30362097934331500, 606395795174882161, 12016899266310773097, 261771941015999635310

Offset: 1

Seiichi Manyama, Aug 15 2022

nonn

Cf. A087906, A356009, A354341.

a087906(n) = n!*sumdiv(n, d, 1/(d*(n/d)!));
a(n) = sum(k=1, n, a087906(k)*binomial(n, k));

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, (exp(x^k)-1)/k)))

my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, log(1-x^k)/k!)))

a(n) = Sum_{k=1..n} A087906(k) * binomial(n,k).
E.g.f.: exp(x) * Sum_{k>0} (exp(x^k) - 1)/k.
E.g.f.: -exp(x) * Sum_{k>0} log(1-x^k)/k!.

A356599 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k!) )^exp(x).

Original entry on oeis.org

1, 1, 5, 25, 159, 1201, 10488, 102901, 1121375, 13406353, 174284898, 2445111373, 36799134584, 591042564425, 10086822013726, 182218681622851, 3472980343846199, 69632877583186121, 1464890891351327598, 32260213678562913097, 742152913359395190170

Offset: 0

Seiichi Manyama, Aug 15 2022

nonn

Cf. A354341, A356409.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k!)^exp(x)))

a354341(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!^d))/(n-k)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354341(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A354341(k) * binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

A354338 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} 1/(d * (k/d)!) )/(n-k)!.

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A356599 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k!) )^exp(x).

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Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula