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

1, 24, 20160, 79833600, 871782912000, 20274183401472000, 861733891296165888000, 60493719168990845337600000, 6526062423950732395020288000000, 1025113885554181044609786839040000000, 224844379201911853600532206127677440000000, 66595307609539060446820030939720014888960000000

Offset: 0

Author

Karol A. Penson, Oct 12 2009

Keywords

• nonn

Comments

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))) ).

This solution may not be unique.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A009120, A061924.

Programs

Magma

[Factorial(4*n) / Factorial(n): n in [0..15]]; // Vincenzo Librandi, May 10 2016

Maple

A166338_list := proc(n) u:=z^(1/4);(cosh(u)+cos(u))/2:series(%,z,n+2):
seq(1/(i!*coeff(%,z,i)),i=0..n) end: A166338_list(9); # Peter Luschny, Jul 12 2012

Mathematica

Table[(4n)!/n!,{n,0,10}] (* Harvey P. Dale, May 30 2015 *)

Formula

G.f.: Sum_{n>=0} a(n)*x^n/(n!)^3 = hypergeom([1/4, 1/2, 3/4], [1, 1], 256*x).

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).

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

From Seiichi Manyama, May 25 2025: (Start)

a(n) = RisingFactorial(n+1,3*n).

a(n) = (3*n)! * [x^(3*n)] 1/(1 - x)^(n+1). (End)