A336634 - OEIS

%I #9 Jul 30 2020 18:25:51

%S 1,1,0,-4,14,-18,-168,1920,-11898,27398,582896,-13028904,183020620,

%T -2061910004,17930433744,-65293856160,-1965585556410,69343044999750,

%U -1519055329884960,26755366818127560,-374375460816570780,2924763867241325220

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

%H Robert Israel, <a href="/A336634/b336634.txt">Table of n, a(n) for n = 0..450</a>

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

%F D-finite with recurrence: n*a(n) = -(3*n^2 - 7*n + 3)*a(n - 1) + (7 - 3*n)*(n - 1)^2*a(n - 2) - (n - 1)^2*(n - 2)^2*a(n - 3). - _Robert Israel_, Jul 30 2020

%p rec:= n*a(n) = -(3*n^2 - 7*n + 3)*a(n - 1) + (7 - 3*n)*(n - 1)^2*a(n - 2) - (n - 1)^2*(n - 2)^2*a(n - 3):

%p f:= gfun:-rectoproc({rec,a(0)=1,a(1)=1,a(2)=0},a(n),remember):

%p map(f, [$0..30]); # _Robert Israel_, Jul 30 2020

%t nmax = 21; CoefficientList[Series[Exp[-x] BesselI[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2

%t Table[(-1)^n n! HypergeometricPFQ[{1/2, -n}, {1, 1}, 4], {n, 0, 21}]

%t Table[Sum[(-1)^(n - k) Binomial[n, k]^2 Binomial[2 k, k] (n - k)!, {k, 0, n}], {n, 0, 21}]

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k)^2 * binomial(2*k,k) * (n-k)!); \\ _Michel Marcus_, Jul 30 2020

%Y Cf. A000984, A009940, A244973, A336293.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Jul 28 2020