cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A166750 a(n) = (A001147(n))^3 = 2^(3*n)*GAMMA(n+1/2)^3/Pi^(3/2).

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%I A166750 #11 Sep 08 2022 08:45:48
%S A166750 1,1,27,3375,1157625,843908625,1123242379875,2467763508585375,
%T A166750 8328701841475640625,40918912147169822390625,
%U A166750 280662818417437811777296875,2599218361363891574869546359375,31624689802714468791437770554515625,494135778167413574866215164914306640625
%N A166750 a(n) = (A001147(n))^3 = 2^(3*n)*GAMMA(n+1/2)^3/Pi^(3/2).
%H A166750 Vincenzo Librandi, <a href="/A166750/b166750.txt">Table of n, a(n) for n = 0..160</a>
%F A166750 G.f.: sum(a(n)*x^n/(n!)^3,n=0..infinity) = 4*EllipticK((1/2)*sqrt(2-2*sqrt(1-8*x)))^2/Pi^2, sum(a(n)*x^n/(n!)^4,n=0..infinity)=hypergeom([1/2,1/2,1/2],[1,1,1],8*x).
%F A166750 Asymptotics: a(n) = (2*sqrt(2)/((exp(-1/2))^3*(exp(1/2))^3)-(1/4)*sqrt(2)/((exp(-1/2))^3*(exp(1/2))^3*n)+(1/64)*sqrt(2)/((exp(-1/2))^3*(exp(1/2))^3*n^2)+O(1/n^3))*(2^n)^3/(((1/n)^n)^3*(exp(n))^3), n->infinity.
%F A166750 Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem),in Maple notation:
%F A166750 a(n) = int(x^n*MeijerG([[],[]],[[ -1/2,-1/2,-1/2],[]],x/8)/(8*(Pi)^(3/2)), x=0..infinity), n=0,1... .
%F A166750 This solution is not unique.
%F A166750 a(n) -(2*n-1)^3*a(n-1) +a(n-2) -(2*n-5)^3*a(n-3) =0. - _R. J. Mathar_, Jul 24 2012
%p A166750 seq((doublefactorial(2*n-1))^3, n=0..15);
%t A166750 Table[((2 n - 1)!!)^3, {n, 0, 30}] (* _Vincenzo Librandi_, Jul 21 2017 *)
%o A166750 (Magma) DoubleFactorial:=func< n | &*[n..2 by -2] >; [DoubleFactorial((2*n-1))^3: n in [0..20]]; // _Vincenzo Librandi_, Jul 21 2017
%K A166750 nonn
%O A166750 0,3
%A A166750 _Karol A. Penson_, Oct 21 2009