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A100733 a(n) = (4*n)!. 10
1, 24, 40320, 479001600, 20922789888000, 2432902008176640000, 620448401733239439360000, 304888344611713860501504000000, 263130836933693530167218012160000000, 371993326789901217467999448150835200000000, 815915283247897734345611269596115894272000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Integral representation of a(n) as n-th moment of a positive function

W(x) on the positive axis, in Maple notation:

a(n)=int(x^n*W(x),x=0..infinity)=int(x^n*(1/4)*exp(-x^(1/4))/x^(3/4),x=0..infinity), n=0,1... .

This is the solution of the Stieltjes moment problem with the moments a(n), n=0,1... .

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

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... (End)

MATHEMATICA

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

PROG

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

CROSSREFS

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

Sequence in context: A188952 A258901 A062322 * A158664 A125048 A003920

Adjacent sequences:  A100730 A100731 A100732 * A100734 A100735 A100736

KEYWORD

nonn,easy

AUTHOR

Ralf Stephan, Dec 08 2004

EXTENSIONS

More terms from Harvey P. Dale, Oct 03 2014

STATUS

approved

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Last modified October 16 08:24 EDT 2018. Contains 316259 sequences. (Running on oeis4.)