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A166338 a(n) = (4*n)!/n!.

%I A166338 #28 May 25 2025 09:20:56
%S A166338 1,24,20160,79833600,871782912000,20274183401472000,
%T A166338 861733891296165888000,60493719168990845337600000,
%U A166338 6526062423950732395020288000000,1025113885554181044609786839040000000,224844379201911853600532206127677440000000,66595307609539060446820030939720014888960000000
%N A166338 a(n) = (4*n)!/n!.
%C A166338 Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation: a(n) = Integral_{x=0..oo} ( x^n*((1/16*(2*Pi^(3/2)*sqrt(2)*hypergeom([], [1/2, 3/4], -(1/256)*x)*sqrt(x) -2*Pi*sqrt(2)*hypergeom([], [3/4, 5/4], -(1/256)*x)*Gamma(3/4)*x^(3/4) +sqrt(Pi)*Gamma(3/4)^2*hypergeom([], [5/4, 3/2], -(1/256)*x)*x))*sqrt(2)/(Gamma(3/4)*x^(5/4)*Pi^(3/2))) ).
%C A166338 This solution may not be unique.
%H A166338 G. C. Greubel, <a href="/A166338/b166338.txt">Table of n, a(n) for n = 0..100</a>
%F A166338 G.f.: Sum_{n>=0} a(n)*x^n/(n!)^3 = hypergeom([1/4, 1/2, 3/4], [1, 1], 256*x).
%F A166338 a(n) ~ 2*(1-1/(16*n)+1/(512*n^2)+331/(122880*n^3)+O(1/n^4)))*(2^n)^8/(((1/n)^n)^3*(exp(n))^3).
%F A166338 1/a(n) = n!*[x^n](cosh(x^(1/4))+cos(x^(1/4)))/2. - _Peter Luschny_, Jul 12 2012
%F A166338 From _Seiichi Manyama_, May 25 2025: (Start)
%F A166338 a(n) = RisingFactorial(n+1,3*n).
%F A166338 a(n) = (3*n)! * [x^(3*n)] 1/(1 - x)^(n+1). (End)
%p A166338 A166338_list := proc(n) u:=z^(1/4);(cosh(u)+cos(u))/2:series(%,z,n+2):
%p A166338 seq(1/(i!*coeff(%,z,i)),i=0..n) end: A166338_list(9); # _Peter Luschny_, Jul 12 2012
%t A166338 Table[(4n)!/n!,{n,0,10}] (* _Harvey P. Dale_, May 30 2015 *)
%o A166338 (Magma) [Factorial(4*n) / Factorial(n): n in [0..15]]; // _Vincenzo Librandi_, May 10 2016
%Y A166338 Cf. A009120, A061924.
%K A166338 nonn
%O A166338 0,2
%A A166338 _Karol A. Penson_, Oct 12 2009