A158094 G.f. Product_{n>=1} (1 + a(n)*x^n) = Sum_{n>=0} n!*x^n.

1, 2, 4, 20, 92, 580, 4156, 34372, 314348, 3204116, 35703996, 433587396, 5687955724, 80265513140, 1211781628060, 19497537309028, 333041104402860, 6019819589363348, 114794574818830716, 2303337794614783236

Offset: 1

Author

Paul D. Hanna, Apr 15 2009

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..450

Crossrefs

Cf. A316084.

Programs

Maple

A158094:= proc(n)
option remember;
local S;
S:= series(add(k!*x^k,k=0..n)/mul(1+A158094(k)*x^k,k=1..n-1),x,n+1);
coeff(S,x,n)
end; # Robert Israel, Mar 04 2014

Mathematica

a[n_] := a[n] = Module[{s}, s = Series[Sum[k!*x^k, {k, 0, n}]/Product[1+a[k]*x^k, {k, 1, n-1}], {x, 0, n+1}]; Coefficient[s, x, n]]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 04 2014, after Maple *)

PARI

{a(n)=if(n<1, 0, polcoeff(sum(k=0,n,k!*x^k)/prod(k=1, n-1, 1+a(k)*x^k +x*O(x^n)), n))}

Formula

a(n) ~ n! * (1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - 171/n^5 - 1542/n^6 - 16241/n^7 - 194973/n^8 - 2622610/n^9 - 39027573/n^10 - ...), for coefficients see A113869. - Vaclav Kotesovec, Jun 18 2019