A291699 a(n) = n^n*(2*n)!/(n!*(n + 1)!).

1, 1, 8, 135, 3584, 131250, 6158592, 353299947, 23991418880, 1883638417518, 167960000000000, 16772331868538246, 1854655886442627072, 225005916687384753700, 29718395534545380311040, 4245313393689422607421875, 652233889532678001886494720, 107247390031799133661006687830

Offset: 0

Author

Ilya Gutkovskiy, Aug 30 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..322

Crossrefs

Main diagonal of A290605.

Cf. A000108, A000312, A001761, A061711.

Programs

Maple

seq(n^n*(2*n)!/n!/(n+1)!, n=0..50); # Robert Israel, Aug 30 2017

Mathematica

Join[{1}, Table[n^n (2 n)!/(n! (n + 1)!), {n, 1, 17}]]
Table[SeriesCoefficient[2/(1 + Sqrt[1 - 4 n x]), {x, 0, n}], {n, 0, 17}]
Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-n x, 1, {i, 1, n}]), {x, 0, n}], {n, 0, 17}]

PARI

a(n)=binomial(2*n,n)/(n+1)*n^n \\ Charles R Greathouse IV, Oct 23 2023

Formula

a(n) = [x^n] 2/(1 + sqrt(1 - 4*n*x)).

a(n) = [x^n] 1/(1 - n*x/(1 - n*x/(1 - n*x/(1 - n*x/(1 - n*x/(1 - ...)))))), a continued fraction.

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

a(n) = n^n*binomial(2*n,n)/(n + 1).

a(n) = A000312(n)*A000108(n).

a(n) = A290605(n,n).

a(n) ~ 4^n*n^(n-3/2)/sqrt(Pi).