A291699 - OEIS

%I #9 Oct 23 2023 12:46:33

%S 1,1,8,135,3584,131250,6158592,353299947,23991418880,1883638417518,

%T 167960000000000,16772331868538246,1854655886442627072,

%U 225005916687384753700,29718395534545380311040,4245313393689422607421875,652233889532678001886494720,107247390031799133661006687830

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

%H Robert Israel, <a href="/A291699/b291699.txt">Table of n, a(n) for n = 0..322</a>

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

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

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

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

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

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

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

%p seq(n^n*(2*n)!/n!/(n+1)!, n=0..50); # _Robert Israel_, Aug 30 2017

%t Join[{1}, Table[n^n (2 n)!/(n! (n + 1)!), {n, 1, 17}]]

%t Table[SeriesCoefficient[2/(1 + Sqrt[1 - 4 n x]), {x, 0, n}], {n, 0, 17}]

%t Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-n x, 1, {i, 1, n}]), {x, 0, n}], {n, 0, 17}]

%o (PARI) a(n)=binomial(2*n,n)/(n+1)*n^n \\ _Charles R Greathouse IV_, Oct 23 2023

%Y Main diagonal of A290605.

%Y Cf. A000108, A000312, A001761, A061711.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 30 2017