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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Magma
[(3^n*Factorial(2*n)/Factorial(n)^2)*(2*n+1): n in [0..20]]; // Vincenzo Librandi, Jul 05 2011
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Mathematica
CoefficientList[Series[(1-12x)^(-3/2),{x,0,20}],x] (* Harvey P. Dale, Oct 26 2016 *)
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)