A294409 a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*n*x).

1, 1, 12, 189, 4864, 159375, 6578496, 323652399, 18572378112, 1216112914971, 89530000000000, 7319100286183983, 657910135976361984, 64494528072860946073, 6847518630093139525632, 782782183702056884765625, 95860848315529046085599232, 12520224284071636768582166787, 1737254440584625641929018966016

Offset: 0

Author

Ilya Gutkovskiy, Oct 30 2017

Keywords

• nonn

Comments

a(n) is the central coefficient of (1 + n*x + n^2*x^2)^n.

Links

• Robert Israel, Table of n, a(n) for n = 0..333

Crossrefs

Cf. A000312, A002426, A098453, A186925, A292629.

Programs

Maple

seq(coeff((1+n*x+n^2*x^2)^n,x,n),n=0..100); # Robert Israel, Oct 30 2017

Mathematica

Table[n! SeriesCoefficient[Exp[n x] BesselI[0, 2 n x], {x, 0, n}], {n, 0, 18}]
Table[CoefficientList[Series[(1 + n x + n^2 x^2)^n, {x, 0, n}], x][[-1]], {n, 0, 18}]
Table[SeriesCoefficient[1/Sqrt[(1 + n x) (1 - 3 n x)], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n^n Sum[Binomial[n, k] Binomial[k, n - k],{k, 0, n}], {n, 1, 18}]]
Join[{1}, Table[n^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {1}, 4], {n, 1, 18}]]

Formula

a(n) = [x^n] 1/sqrt((1 + n*x)*(1 - 3*n*x)).

a(n) = A000312(n)*A002426(n).

a(n) ~ sqrt(3)*3^n*n^n/(2*sqrt(Pi*n)).