A356486 -id:A356486 - cp's OEIS frontend

A356487 Expansion of e.g.f. Product_{k>0} 1/(1 - (k * x)^k)^(1/k!).

1, 1, 6, 45, 580, 7105, 170076, 2654575, 116426528, 2386183761, 209503380160, 3455683548691, 969334978024920, 15164681616944353, 6510178188269825720, 223847763757748796975, 81261936394687862700256, 1581790511799886415713825

Offset: 0

Seiichi Manyama, Aug 09 2022

nonn

Cf. A023882, A209902, A356486.

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

a356486(n) = (n-1)!*sumdiv(n, d, d^n/(d-1)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356486(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A356486(k) * binomial(n-1,k-1) * a(n-k).

A356529 a(n) = (n-1)! * Sum_{d|n} d^(n-d+1).

Original entry on oeis.org

1, 3, 8, 78, 144, 14400, 5760, 5851440, 88583040, 5859786240, 43545600, 24077414592000, 6706022400, 35948640894566400, 4395744249613516800, 263312496059348736000, 376610217984000, 5901087844517892009984000, 128047474114560000

Offset: 1

Seiichi Manyama, Aug 10 2022

nonn

Cf. A342675, A356486, A356530.

a[n_] := (n - 1)! * DivisorSum[n, #^(n - # + 1) &]; Array[a, 20] (* Amiram Eldar, Aug 10 2022 *)

PARI
```
a(n) = (n-1)!*sumdiv(n, d, d^(n-d+1));
```

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

a(n) = (n-1)! * A342675(n).
If p is prime, a(p) = (1+p) * (p-1)!.
E.g.f.: -Sum_{k>0} log(1 - (k * x)^k)/k^k.

cp's OEIS Frontend

A356487 Expansion of e.g.f. Product_{k>0} 1/(1 - (k * x)^k)^(1/k!).

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A356529 a(n) = (n-1)! * Sum_{d|n} d^(n-d+1).

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