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

Offset

0,3

Links

Table of n, a(n) for n=0..17.

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). [From 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

Example

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! +...

Maple

a:=n->sum(stirling2(n, j)*n!/j!, j=0..n):seq(a(n), n=0..15); # Zerinvary Lajos, Mar 19 2007

Mathematica

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

Prog

(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 */

Crossrefs

Cf. A001569.

Sequence in context: A141628 A048802 A213357 * A307979 A129043 A182951

Adjacent sequences: A119389 A119390 A119391 * A119393 A119394 A119395

Keyword

easy,nonn

Author

Vladeta Jovovic, Jul 25 2006

Extensions

More terms from Stefan Steinerberger, Nov 23 2007

Status

approved