A115903 Expansion of (1-12*x)^(-3/2).

1, 18, 270, 3780, 51030, 673596, 8756748, 112586760, 1435481190, 18182761740, 229102797924, 2874198737592, 35927484219900, 447711726432600, 5564417171376600, 68998772925069840, 853859814947739270, 10547680067001485100, 130088054159684982900, 1602137088071909789400

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f.: 1/((1-12*x)*sqrt(1-12*x)).

a(n) = Jacobi_P(n,1/2,1/2,1)*12^n.

a(n) = 3^n*(2*n+1)*binomial(2*n,n) = 3^n*A002457(n).

a(n) = (2*n+1)*A098658(n).

D-finite with recurrence: n*a(n) - 6*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 07 2012

From Amiram Eldar, Jan 27 2024: (Start)

Sum_{n>=0} 1/a(n) = 12*arcsin(1/sqrt(12))/sqrt(11).

Sum_{n>=0} (-1)^n/a(n) = 12*arcsinh(1/sqrt(12))/sqrt(13). (End)

Mathematica

CoefficientList[Series[(1-12x)^(-3/2), {x, 0, 20}], x] (* Harvey P. Dale, Oct 26 2016 *)

Prog

(Magma) [(3^n*Factorial(2*n)/Factorial(n)^2)*(2*n+1): n in [0..20]]; // Vincenzo Librandi, Jul 05 2011

Crossrefs

Cf. A002457, A098658.

Sequence in context: A076693 A083445 A159740 * A004357 A249598 A245924

Adjacent sequences: A115900 A115901 A115902 * A115904 A115905 A115906

Keyword

easy,nonn

Author

Paul Barry, Feb 02 2006

Status

approved