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
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 0..450
Programs
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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}] -
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
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