A349468 - OEIS

%I #11 Mar 06 2022 08:31:12

%S 1,12,840,110880,21621600,5587021440,1799020903680,693908062848000,

%T 311911674250176000,160114659448423680000,92418181433630148096000,

%U 59248455951814527670272000,41770161446029242007541760000,32118041062654484854414417920000,26749739913610806671605150924800000

%N a(n) = (4*n)! / (n! * (2*n)!).

%F E.g.f.: 2 * EllipticK( 16*sqrt(x) / (1 + 8*sqrt(x)) ) / (Pi * sqrt(1 + 8*sqrt(x))).

%F a(n) is the coefficient of x^n in expansion of d^n/dx^n g(x), where g(x) is the g.f. of central binomial coefficients (A000984).

%F a(n) = n! * A000897(n) = A009120(n) / n! = A166338(n) / (2*n)! = A001448(n) * A001813(n).

%F a(n) ~ 64^n * n^(n-1/2) / (sqrt(Pi) * exp(n)).

%F D-finite with recurrence n*a(n) -4*(4*n-1)*(4*n-3)*a(n-1)=0. - _R. J. Mathar_, Mar 06 2022

%t Table[(4 n)!/(n! (2 n)!), {n, 0, 14}]

%t nmax = 14; CoefficientList[Series[2 EllipticK[16 Sqrt[x]/(1 + 8 Sqrt[x])]/(Pi Sqrt[1 + 8 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[SeriesCoefficient[D[1/Sqrt[1 - 4 x], {x, n}], {x, 0, n}], {n, 0, 14}]

%o (PARI) a(n) = (4*n)! / (n! * (2*n)!) \\ _Andrew Howroyd_, Nov 20 2021

%Y Cf. A000897, A000984, A001448, A001813, A009120, A166338.

%K nonn,easy

%O 0,2

%A _Ilya Gutkovskiy_, Nov 18 2021