A336634 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(-x) * BesselI(0,2*sqrt(x))^2.

1, 1, 0, -4, 14, -18, -168, 1920, -11898, 27398, 582896, -13028904, 183020620, -2061910004, 17930433744, -65293856160, -1965585556410, 69343044999750, -1519055329884960, 26755366818127560, -374375460816570780, 2924763867241325220

Offset

0,4

Links

Robert Israel, Table of n, a(n) for n = 0..450

Formula

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

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

Maple

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):
f:= gfun:-rectoproc({rec, a(0)=1, a(1)=1, a(2)=0}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jul 30 2020

Mathematica

nmax = 21; CoefficientList[Series[Exp[-x] BesselI[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2
Table[(-1)^n n! HypergeometricPFQ[{1/2, -n}, {1, 1}, 4], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k]^2 Binomial[2 k, k] (n - k)!, {k, 0, n}], {n, 0, 21}]

Prog

(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

Crossrefs

Cf. A000984, A009940, A244973, A336293.

Sequence in context: A032825 A251966 A022383 * A045248 A271375 A070902

Adjacent sequences: A336631 A336632 A336633 * A336635 A336636 A336637

Keyword

sign

Author

Ilya Gutkovskiy, Jul 28 2020

Status

approved