A352914 Expansion of e.g.f. exp(Sum_{k>=1} prime(k)*x^k).

1, 2, 10, 74, 676, 7592, 97024, 1416200, 23015248, 412777952, 8090869984, 171435904928, 3908548404160, 95264270043776, 2470715015425024, 67913132377486208, 1971038886452490496, 60212661838223997440, 1930529043247940342272, 64801071784954698480128

Offset: 0

Author

Seiichi Manyama, Apr 28 2022

Keywords

• nonn

Crossrefs

Cf. A000040, A007446, A033286, A353162, A353166.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      ithprime(j)*j*binomial(n, j)*j!, j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 28 2022

Mathematica

a[0] = 1; a[n_] := a[n] = (n-1)! Sum[k Prime[k] a[n-k]/(n-k)!, {k, 1, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 28 2022 *)

PARI

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

PARI

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*prime(k)*a(n-k)/(n-k)!));

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A033286(k) * a(n-k)/(n-k)!.