A119392 a(n) = n! * Sum_{k=0..n} Stirling2(n,k)/k!.

1, 1, 3, 16, 133, 1571, 24721, 496168, 12317761, 369451477, 13135552831, 545021905176, 26051269951213, 1418976050686351, 87262518335077541, 6010361475663954256, 460405692649973927041, 38981134670714611635913, 3627857520994811984369371, 369309424376334817020139840

Offset: 0

Author

Vladeta Jovovic, Jul 25 2006

Keywords

• easy
• nonn

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 133*x^4/4! +...
where A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-2*x))/2! + x^3/((1-x)*(1-2*x)*(1-3*x))/3! +...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A001569.

Programs

Maple

a:=n->sum(Stirling2(n,j)*n!/j!,j=0..n):seq(a(n),n=0..15); # Zerinvary Lajos, Mar 19 2007
# second Maple program:
b:= proc(n, k) option remember;
     `if`(n=0, 1/k!, k*b(n-1, k)+b(n-1, k+1))
    end:
a:= n-> n!*b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2025

Mathematica

Table[n!*Sum[StirlingS2[n, k]/k!, {k, 0, n}], {n, 0, 20}] (* Stefan Steinerberger, Nov 23 2007 *)

PARI

{a(n)=n!*polcoeff(sum(m=0,n,x^m/m!/prod(k=1,m,1-k*x +x*O(x^n))),n)} /* Paul D. Hanna */

Formula

Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0,2*sqrt(exp(x)-1)).

E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} 1/(1-k*x). - Paul D. Hanna, Dec 13 2011

E.g.f.: 1 + x*(1 - E(0) )/(1-x) where E(k) = 1 - 1/(1-x*(k+1))/(k+1)/(1-x/(x-1/E(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 18 2013

Extensions

More terms from Stefan Steinerberger, Nov 23 2007