cp's OEIS Frontend

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

%I A100733 #36 May 02 2025 07:58:40
%S A100733 1,24,40320,479001600,20922789888000,2432902008176640000,
%T A100733 620448401733239439360000,304888344611713860501504000000,
%U A100733 263130836933693530167218012160000000,371993326789901217467999448150835200000000,815915283247897734345611269596115894272000000000
%N A100733 a(n) = (4*n)!.
%C A100733 From _Karol A. Penson_, Jun 11 2009: (Start)
%C A100733 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 A100733 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 A100733 This is the solution of the Stieltjes moment problem with the moments a(n), n >= 0.
%C A100733 As the moments a(n) grow very rapidly this suggests, but does not prove, that this solution may not be unique.
%C A100733 This is indeed the case as by construction the following "doubly" infinite family:
%C A100733 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 A100733 with the restrictions k=+-1,+-2,..., abs(a) < 1 is still positive on 0 <= x < infinity and has moments a(n).
%C A100733 (End)
%H A100733 Vincenzo Librandi, <a href="/A100733/b100733.txt">Table of n, a(n) for n = 0..100</a>
%F A100733 E.g.f.: 1/(1-x^4).
%F A100733 From _Ilya Gutkovskiy_, Jan 20 2017: (Start)
%F A100733 a(n) ~ sqrt(Pi)*2^(8*n+3/2)*n^(4*n+1/2)/exp(4*n).
%F A100733 Sum_{n>=0} 1/a(n) = (cos(1) + cosh(1))/2 = 1.04169147034169174... = A332890. (End)
%F A100733 Sum_{n>=0} (-1)^n/a(n) = cos(1/sqrt(2))*cosh(1/sqrt(2)). - _Amiram Eldar_, Feb 14 2021
%t A100733 (4*Range[0,20])! (* _Harvey P. Dale_, Oct 03 2014 *)
%o A100733 (Magma) [Factorial(4*n): n in [0..10]]; // _Vincenzo Librandi_, Sep 24 2011
%Y A100733 Cf. A000142, A010050, A100732, A100734, A268505, A332890.
%K A100733 nonn,easy
%O A100733 0,2
%A A100733 _Ralf Stephan_, Dec 08 2004
%E A100733 More terms from _Harvey P. Dale_, Oct 03 2014