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

%I A115903 #20 Jan 27 2024 10:29:58
%S A115903 1,18,270,3780,51030,673596,8756748,112586760,1435481190,18182761740,
%T A115903 229102797924,2874198737592,35927484219900,447711726432600,
%U A115903 5564417171376600,68998772925069840,853859814947739270,10547680067001485100,130088054159684982900,1602137088071909789400
%N A115903 Expansion of (1-12*x)^(-3/2).
%H A115903 Vincenzo Librandi, <a href="/A115903/b115903.txt">Table of n, a(n) for n = 0..200</a>
%F A115903 G.f.: 1/((1-12*x)*sqrt(1-12*x)).
%F A115903 a(n) = Jacobi_P(n,1/2,1/2,1)*12^n.
%F A115903 a(n) = 3^n*(2*n+1)*binomial(2*n,n) = 3^n*A002457(n).
%F A115903 a(n) = (2*n+1)*A098658(n).
%F A115903 D-finite with recurrence: n*a(n) - 6*(2*n+1)*a(n-1) = 0. - _R. J. Mathar_, Nov 07 2012
%F A115903 From _Amiram Eldar_, Jan 27 2024: (Start)
%F A115903 Sum_{n>=0} 1/a(n) = 12*arcsin(1/sqrt(12))/sqrt(11).
%F A115903 Sum_{n>=0} (-1)^n/a(n) = 12*arcsinh(1/sqrt(12))/sqrt(13). (End)
%t A115903 CoefficientList[Series[(1-12x)^(-3/2),{x,0,20}],x] (* _Harvey P. Dale_, Oct 26 2016 *)
%o A115903 (Magma) [(3^n*Factorial(2*n)/Factorial(n)^2)*(2*n+1): n in [0..20]]; // _Vincenzo Librandi_, Jul 05 2011
%Y A115903 Cf. A002457, A098658.
%K A115903 easy,nonn
%O A115903 0,2
%A A115903 _Paul Barry_, Feb 02 2006