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%I #10 Oct 30 2017 22:01:57
%S 1,1,12,189,4864,159375,6578496,323652399,18572378112,1216112914971,
%T 89530000000000,7319100286183983,657910135976361984,
%U 64494528072860946073,6847518630093139525632,782782183702056884765625,95860848315529046085599232,12520224284071636768582166787,1737254440584625641929018966016
%N a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*n*x).
%C a(n) is the central coefficient of (1 + n*x + n^2*x^2)^n.
%H Robert Israel, <a href="/A294409/b294409.txt">Table of n, a(n) for n = 0..333</a>
%F a(n) = [x^n] 1/sqrt((1 + n*x)*(1 - 3*n*x)).
%F a(n) = A000312(n)*A002426(n).
%F a(n) ~ sqrt(3)*3^n*n^n/(2*sqrt(Pi*n)).
%p seq(coeff((1+n*x+n^2*x^2)^n,x,n),n=0..100); # _Robert Israel_, Oct 30 2017
%t Table[n! SeriesCoefficient[Exp[n x] BesselI[0, 2 n x], {x, 0, n}], {n, 0, 18}]
%t Table[CoefficientList[Series[(1 + n x + n^2 x^2)^n, {x, 0, n}], x][[-1]], {n, 0, 18}]
%t Table[SeriesCoefficient[1/Sqrt[(1 + n x) (1 - 3 n x)], {x, 0, n}], {n, 0, 18}]
%t Join[{1}, Table[n^n Sum[Binomial[n, k] Binomial[k, n - k],{k, 0, n}], {n, 1, 18}]]
%t Join[{1}, Table[n^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {1}, 4], {n, 1, 18}]]
%Y Cf. A000312, A002426, A098453, A186925, A292629.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Oct 30 2017
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