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 A294409 a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*n*x). 1
 1, 1, 12, 189, 4864, 159375, 6578496, 323652399, 18572378112, 1216112914971, 89530000000000, 7319100286183983, 657910135976361984, 64494528072860946073, 6847518630093139525632, 782782183702056884765625, 95860848315529046085599232, 12520224284071636768582166787, 1737254440584625641929018966016 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 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)). 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}]] CROSSREFS Cf. A000312, A002426, A098453, A186925, A292629. Sequence in context: A285410 A218886 A239294 * A239776 A071990 A230757 Adjacent sequences: A294406 A294407 A294408 * A294410 A294411 A294412 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Oct 30 2017 STATUS approved

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Last modified March 21 16:33 EDT 2023. Contains 361408 sequences. (Running on oeis4.)