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A119399 a(n) = Sum_{k=0..n} (n!/k!)^2*binomial(n-1,k-1). 1

%I

%S 1,1,5,55,1057,31301,1319581,74996755,5521809665,510921831817,

%T 58003632177301,7924389193344911,1282139184447959905,

%U 242395881776602480525,52937407769332221775277,13223898129391280722348651,3746106716895295870279280641,1194375522748111467993501362705

%N a(n) = Sum_{k=0..n} (n!/k!)^2*binomial(n-1,k-1).

%H Andrew Howroyd, <a href="/A119399/b119399.txt">Table of n, a(n) for n = 0..100</a>

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

%F Special values of hypergeometric function of type 1F2. In Maple notation: a(n)=((n!)^2)*hypergeom([1-n],[2,2],-1), n=0,1... . This sequence arises in exponentiating the operator D=d(x^2)(d^2), where d=d/dx. - _Karol A. Penson_, Nov 22 2008

%F Recurrence: a(n) = (3*n^2-5*n+3)*a(n-1) + (n-3)*(n-1)^2*(n-2)^3*a(n-3) - (n-1)^2*(3*n-4)*(n-2)*a(n-2). - _Vaclav Kotesovec_, Jun 03 2013

%F a(n) ~ n^(2*n)*exp(3*n^(1/3)-2*n)/sqrt(3). - _Vaclav Kotesovec_, Jun 03 2013

%t CoefficientList[Series[BesselI[0,2*Sqrt[x/(1-x)]], {x, 0, 20}], x]* Range[0, 20]!^2 (* _Vaclav Kotesovec_, Jun 03 2013 *)

%o (PARI) a(n)={if(n<1, n==0, sum(k=0, n, (n!/k!)^2*binomial(n-1,k-1)))} \\ _Andrew Howroyd_, Jan 08 2020

%K easy,nonn

%O 0,3

%A _Vladeta Jovovic_, Jul 25 2006

%E Terms a(15) and beyond from _Andrew Howroyd_, Jan 08 2020

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Last modified September 22 19:36 EDT 2021. Contains 347608 sequences. (Running on oeis4.)