A349468 a(n) = (4*n)! / (n! * (2*n)!).

1, 12, 840, 110880, 21621600, 5587021440, 1799020903680, 693908062848000, 311911674250176000, 160114659448423680000, 92418181433630148096000, 59248455951814527670272000, 41770161446029242007541760000, 32118041062654484854414417920000, 26749739913610806671605150924800000

Offset

0,2

Links

Table of n, a(n) for n=0..14.

Formula

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

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).

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

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

D-finite with recurrence n*a(n) -4*(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Mar 06 2022

Mathematica

Table[(4 n)!/(n! (2 n)!), {n, 0, 14}]
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]!
Table[SeriesCoefficient[D[1/Sqrt[1 - 4 x], {x, n}], {x, 0, n}], {n, 0, 14}]

Prog

(PARI) a(n) = (4*n)! / (n! * (2*n)!) \\ Andrew Howroyd, Nov 20 2021

Crossrefs

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

Sequence in context: A280333 A350412 A207817 * A203410 A275568 A271433

Adjacent sequences: A349465 A349466 A349467 * A349469 A349470 A349471

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Nov 18 2021

Status

approved