A351433 a(n) = n! * [x^n] 1/(1 + f^n(x)), where f(x) = exp(x) - 1.

1, -1, 0, 0, -2, -75, -3334, -192864, -14443260, -1372372623, -162009663365, -23314158802286, -4022712394579207, -820399656345934444, -195326656416326556562, -53709209673236813446542, -16896296201917398543629108, -6030879950631118091070849321

Offset: 0

Author

Seiichi Manyama, Feb 11 2022

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..247

Crossrefs

Main diagonal of A351429.

Cf. A139383, A302358, A351424.

Programs

Maple

g:= x-> exp(x)-1:
a:= n-> n! * coeff(series(1/(1+(g@@n)(x)), x, n+1), x, n):
seq(a(n), n=0..22);  # Alois P. Heinz, Feb 11 2022

Mathematica

T[n_, 0] := (-1)^n * n!; T[n_, k_] := T[n, k] = Sum[StirlingS2[n, j]*T[j, k - 1], {j, 0, n}]; a[n_] := T[n, n]; Array[a, 17, 0] (* Amiram Eldar, Feb 11 2022 *)

PARI

T(n, k) = if(k==0, (-1)^n*n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
a(n) = T(n, n);

Formula

a(n) = T(n,n), T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = (-1)^n * n!.