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%I #9 Aug 09 2015 15:35:36
%S 1,1,-3,19,-191,2301,-5579,-2972633,365848449,-41439009671,
%T 5100344009101,-707810961855909,111655250271582337,
%U -19997759486622720971,4047974925567723953349,-920668079777059041167249,233796999474238422487503361,-65865180249832257997559536143
%N a(n) = Sum_{k=0..n} (-1)^(n-k)*(n!/k!)^2*binomial(n-1,k-1).
%F Sum_{>=0} a(n)*x^n/n!^2 = BesselI(0,2*sqrt(x/(1+x))).
%F Recurrence: a(n) = -(3*n^2 - 5*n + 1)*a(n-1) - (n-2)*(n-1)^2*(3*n-4)*a(n-2) - (n-3)*(n-2)^3*(n-1)^2*a(n-3). - _Vaclav Kotesovec_, Mar 02 2014
%p A119394 := proc(n) local k ; add((-1)^(n-k)*(n!/k!)^2*binomial(n-1,k-1),k=0..n) ; end: seq(A119394(n),n=0..20) ; # _R. J. Mathar_, Jan 21 2008
%t Table[Sum[(-1)^(n-k)*(n!/k!)^2*Binomial[n-1,k-1],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Mar 02 2014 *)
%K easy,sign
%O 0,3
%A _Vladeta Jovovic_, Jul 25 2006
%E More terms from _R. J. Mathar_, Jan 21 2008