login
a(n) = (4*n)!/n!.
6

%I #18 Sep 08 2022 08:45:48

%S 1,24,20160,79833600,871782912000,20274183401472000,

%T 861733891296165888000,60493719168990845337600000,

%U 6526062423950732395020288000000,1025113885554181044609786839040000000

%N a(n) = (4*n)!/n!.

%C 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)=int(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))), x=0..infinity), n=0,1... .

%C This solution may not be unique.

%H G. C. Greubel, <a href="/A166338/b166338.txt">Table of n, a(n) for n = 0..100</a>

%F G.f.: sum(a(n)*x^n/(n!)^3,n=0..infinity)=hypergeom([1/4, 1/2, 3/4], [1, 1], 256*x).

%F 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),n->infinity.

%F 1/a(n) = n!*[x^n](cosh(x^(1/4))+cos(x^(1/4)))/2. - _Peter Luschny_, Jul 12 2012

%p A166338_list := proc(n) u:=z^(1/4);(cosh(u)+cos(u))/2:series(%,z,n+2):

%p seq(1/(i!*coeff(%,z,i)),i=0..n) end: A166338_list(9); # _Peter Luschny_, Jul 12 2012

%t Table[(4n)!/n!,{n,0,10}] (* _Harvey P. Dale_, May 30 2015 *)

%o (Magma) [Factorial(4*n) / Factorial(n): n in [0..15]]; // _Vincenzo Librandi_, May 10 2016

%K nonn

%O 0,2

%A _Karol A. Penson_, Oct 12 2009