cp's OEIS Frontend

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

%I A119392 #22 Aug 04 2025 10:10:47
%S A119392 1,1,3,16,133,1571,24721,496168,12317761,369451477,13135552831,
%T A119392 545021905176,26051269951213,1418976050686351,87262518335077541,
%U A119392 6010361475663954256,460405692649973927041,38981134670714611635913,3627857520994811984369371,369309424376334817020139840
%N A119392 a(n) = n! * Sum_{k=0..n} Stirling2(n,k)/k!.
%H A119392 Alois P. Heinz, <a href="/A119392/b119392.txt">Table of n, a(n) for n = 0..300</a>
%F A119392 Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0,2*sqrt(exp(x)-1)).
%F A119392 E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} 1/(1-k*x). - _Paul D. Hanna_, Dec 13 2011
%F A119392 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
%e A119392 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 133*x^4/4! +...
%e A119392 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! +...
%p A119392 a:=n->sum(Stirling2(n,j)*n!/j!,j=0..n):seq(a(n),n=0..15); # _Zerinvary Lajos_, Mar 19 2007
%p A119392 # second Maple program:
%p A119392 b:= proc(n, k) option remember;
%p A119392      `if`(n=0, 1/k!, k*b(n-1, k)+b(n-1, k+1))
%p A119392     end:
%p A119392 a:= n-> n!*b(n, 0):
%p A119392 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 04 2025
%t A119392 Table[n!*Sum[StirlingS2[n, k]/k!, {k, 0, n}], {n, 0, 20}] (* _Stefan Steinerberger_, Nov 23 2007 *)
%o A119392 (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_ */
%Y A119392 Cf. A001569.
%K A119392 easy,nonn
%O A119392 0,3
%A A119392 _Vladeta Jovovic_, Jul 25 2006
%E A119392 More terms from _Stefan Steinerberger_, Nov 23 2007