A336293 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2*k,k) * (n-k)!.

1, 3, 16, 116, 1038, 10922, 131256, 1766592, 26253702, 426173906, 7492270416, 141661870088, 2864168171596, 61621248390756, 1404853103594128, 33815954626749600, 856680253728250950, 22777071459869216850, 633968368216974945600, 18430976777427663053400

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

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

a(n) ~ n^n * exp(4*sqrt(n) - n - 2) / sqrt(8*Pi) * (1 + 55/(24*sqrt(n))). - Vaclav Kotesovec, Aug 04 2022

Recurrence: n*a(n) = (3*n^2 + n - 1)*a(n-1) - (n-1)^2*(3*n + 1)*a(n-2) + (n-2)^2*(n-1)^2*a(n-3). - Vaclav Kotesovec, Aug 04 2022

Mathematica

Table[Sum[Binomial[n, k]^2 Binomial[2 k, k] (n - k)!, {k, 0, n}], {n, 0, 19}]
Table[n! HypergeometricPFQ[{1/2, -n}, {1, 1}, -4], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Exp[x] BesselI[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)^2 * binomial(2*k, k) * (n-k)!); \\ Michel Marcus, Jul 17 2020

Crossrefs

Cf. A000984, A002720, A002893.

Sequence in context: A011818 A036248 A111555 * A355718 A221409 A074522

Adjacent sequences: A336290 A336291 A336292 * A336294 A336295 A336296

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 16 2020

Status

approved