A321521 -id:A321521 - cp's OEIS frontend

A321522 Expansion of Product_{k>=1} (1 + x^k)^((k-1)!).

1, 1, 1, 3, 8, 32, 153, 883, 5980, 46660, 411861, 4057263, 44104688, 524243696, 6762188285, 94055795999, 1403061499362, 22342571084082, 378257158227079, 6783952072695685, 128481050502464062, 2562250926987454694, 53668572808754641369, 1177957644341460946099

Offset: 0

Ilya Gutkovskiy, Nov 12 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000142, A051295, A104150, A179327, A261052, A321521.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial((i-1)!, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 10 2021

nmax = 23; CoefficientList[Series[Product[(1 + x^k)^((k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 23}]

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d! ) * x^k/k).

a(n) ~ (n-1)! * (1 + 1/n + 2/n^2 + 7/n^3 + 34/n^4 + 203/n^5 + 1455/n^6 + 12343/n^7 + 121636/n^8 + 1368647/n^9 + 17343274/n^10 + ...). - Vaclav Kotesovec, Nov 13 2018

A358279 a(n) = Sum_{d|n} (d-1)! * d^(n/d).

Original entry on oeis.org

1, 3, 7, 29, 121, 747, 5041, 40433, 362935, 3629433, 39916801, 479006531, 6227020801, 87178326609, 1307674371487, 20922790212353, 355687428096001, 6402373709021811, 121645100408832001, 2432902008212950169, 51090942171709691335, 1124000727778046766849

Offset: 1

Seiichi Manyama, Nov 08 2022

Cf. A038507, A062363, A078308, A217576, A321521, A358280.

a[n_] := DivisorSum[n, (# - 1)! * #^(n/#) &]; Array[a, 22] (* Amiram Eldar, Aug 30 2023 *)

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

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

G.f.: Sum_{k>0} k! * x^k/(1 - k * x^k).

If p is prime, a(p) = 1 + p!.

cp's OEIS Frontend

A321522 Expansion of Product_{k>=1} (1 + x^k)^((k-1)!).

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