A100733 a(n) = (4*n)!.

1, 24, 40320, 479001600, 20922789888000, 2432902008176640000, 620448401733239439360000, 304888344611713860501504000000, 263130836933693530167218012160000000, 371993326789901217467999448150835200000000, 815915283247897734345611269596115894272000000000

Offset: 0

Author

Ralf Stephan, Dec 08 2004

Keywords

• nonn
• easy

Comments

From Karol A. Penson, Jun 11 2009: (Start)

Integral representation of a(n) as n-th moment of a positive function W(x) = (1/4)*exp(-x^(1/4))/x^(3/4) on the positive axis:

a(n) = Integral_{x=0..oo} x^n*W(x) dx = Integral_{x=0..oo} x^n*(1/4)*exp(-x^(1/4))/x^(3/4) dx, n >= 0.

This is the solution of the Stieltjes moment problem with the moments a(n), n >= 0.

As the moments a(n) grow very rapidly this suggests, but does not prove, that this solution may not be unique.

This is indeed the case as by construction the following "doubly" infinite family:

V(k,a,x) = (1/4)*exp(-x^(1/4))*(a*sin((3/4)*Pi*k + tan((1/4)*Pi*k)*x^(1/4)) + 1)/x^(3/4),

with the restrictions k=+-1,+-2,..., abs(a) < 1 is still positive on 0 <= x < infinity and has moments a(n).

(End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000142, A010050, A100732, A100734, A268505, A332890.

Programs

Magma

[Factorial(4*n): n in [0..10]]; // Vincenzo Librandi, Sep 24 2011

Mathematica

(4*Range[0,20])! (* Harvey P. Dale, Oct 03 2014 *)

Formula

E.g.f.: 1/(1-x^4).

From Ilya Gutkovskiy, Jan 20 2017: (Start)

a(n) ~ sqrt(Pi)*2^(8*n+3/2)*n^(4*n+1/2)/exp(4*n).

Sum_{n>=0} 1/a(n) = (cos(1) + cosh(1))/2 = 1.04169147034169174... = A332890. (End)

Sum_{n>=0} (-1)^n/a(n) = cos(1/sqrt(2))*cosh(1/sqrt(2)). - Amiram Eldar, Feb 14 2021

Extensions

More terms from Harvey P. Dale, Oct 03 2014