A100733 - OEIS

%I #36 May 02 2025 07:58:40

%S 1,24,40320,479001600,20922789888000,2432902008176640000,

%T 620448401733239439360000,304888344611713860501504000000,

%U 263130836933693530167218012160000000,371993326789901217467999448150835200000000,815915283247897734345611269596115894272000000000

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

%C From _Karol A. Penson_, Jun 11 2009: (Start)

%C 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:

%C 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.

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

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

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

%C 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),

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

%C (End)

%H Vincenzo Librandi, <a href="/A100733/b100733.txt">Table of n, a(n) for n = 0..100</a>

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

%F From _Ilya Gutkovskiy_, Jan 20 2017: (Start)

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

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

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

%t (4*Range[0,20])! (* _Harvey P. Dale_, Oct 03 2014 *)

%o (Magma) [Factorial(4*n): n in [0..10]]; // _Vincenzo Librandi_, Sep 24 2011

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

%K nonn,easy

%O 0,2

%A _Ralf Stephan_, Dec 08 2004

%E More terms from _Harvey P. Dale_, Oct 03 2014