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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..100

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

Adjacent sequences:  A119396 A119397 A119398 * A119400 A119401 A119402

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Jul 25 2006

EXTENSIONS

Terms a(15) and beyond from Andrew Howroyd, Jan 08 2020

STATUS

approved

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Last modified July 23 22:20 EDT 2021. Contains 346265 sequences. (Running on oeis4.)