A336217 a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n,k)^2 * a(k).

1, 2, 18, 362, 12946, 723402, 58208490, 6375093258, 911949196434, 165104835435146, 36903191037412618, 9980525774650881738, 3212329170232153022314, 1213419234370490738427722, 531582989226188067128503722, 267336170027296964096123899962

Offset: 0

Author

Ilya Gutkovskiy, Jul 12 2020

Keywords

• nonn

Crossrefs

Cf. A004123, A102221.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
nmax = 15; CoefficientList[Series[1/(1 - 2 Sum[x^k/(k!)^2, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

a(n) = (n!)^2 * [x^n] 1 / (1 - 2 * Sum_{k>=1} x^k / (k!)^2).

a(n) = (n!)^2 * [x^n] 1 / (3 - 2 * BesselI(0,2*sqrt(x))).

a(n) ~ (n!)^2 / (2 * BesselI(1, 2*sqrt(r)) * r^(n + 1/2)), where r = 0.4473998881770456142157108538567782213913712561... is the root of the equation 2*BesselI(0, 2*sqrt(r)) = 3. - Vaclav Kotesovec, Jul 17 2020