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A119399
a(n) = Sum_{k=0..n} (n!/k!)^2*binomial(n-1,k-1).
1
1, 1, 5, 55, 1057, 31301, 1319581, 74996755, 5521809665, 510921831817, 58003632177301, 7924389193344911, 1282139184447959905, 242395881776602480525, 52937407769332221775277, 13223898129391280722348651, 3746106716895295870279280641, 1194375522748111467993501362705
OFFSET
0,3
LINKS
FORMULA
Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0,2*sqrt(x/(1-x))).
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
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
a(n) ~ n^(2*n)*exp(3*n^(1/3)-2*n)/sqrt(3). - Vaclav Kotesovec, Jun 03 2013
MATHEMATICA
CoefficientList[Series[BesselI[0, 2*Sqrt[x/(1-x)]], {x, 0, 20}], x]* Range[0, 20]!^2 (* Vaclav Kotesovec, Jun 03 2013 *)
PROG
(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
CROSSREFS
Sequence in context: A300589 A130031 A336289 * A177557 A158690 A280573
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Jul 25 2006
EXTENSIONS
Terms a(15) and beyond from Andrew Howroyd, Jan 08 2020
STATUS
approved