cp's OEIS Frontend

A072020 Sum of an infinite series: a(n) = Sum_{k = 0..infinity} ((1/27) * (3^n)^3 * Gamma(n+1/3*k+1/3) * Gamma(n+1/3*k+2/3) * Gamma(n+1/3*k+1)) / (Gamma(4/3+1/3*k) * Gamma(5/3+1/3*k) * Gamma(2+1/3*k) * exp(1) * k!).

%I A072020 #24 May 02 2025 07:58:35
%S A072020 1,229,207775,472630861,2148321709561,17028146983530961,
%T A072020 214877019857456672479,4044349155369603186936985,
%U A072020 108105412214943249140163409201,3949854849387058592656207156530781,191308664212963089686669131219301608831
%N A072020 Sum of an infinite series: a(n) = Sum_{k = 0..infinity} ((1/27) * (3^n)^3 * Gamma(n+1/3*k+1/3) * Gamma(n+1/3*k+2/3) * Gamma(n+1/3*k+1)) / (Gamma(4/3+1/3*k) * Gamma(5/3+1/3*k) * Gamma(2+1/3*k) * exp(1) * k!).
%H A072020 Sean A. Irvine, <a href="/A072020/b072020.txt">Table of n, a(n) for n = 1..25</a>
%H A072020 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3.
%F A072020 Representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*(exp(-x^(1/3))*BesselI(3, 2*x^(1/6))/(3*exp(1)*x^(7/6))) dx, n >= 1. This representation is unique.
%e A072020 a(2) = 3!*LaguerreL(3, 3,-1) = 229, special value of associated Laguerre polynomial.
%t A072020 a[n_] := Sum[ 1/27*(3^n)^3 * Gamma[n + 1/3*k + 1/3] * Gamma[n + 1/3*k + 2/3] * Gamma[n + 1/3*k + 1] / Gamma[ 4/3 + 1/3*k] / Gamma[5/3 + 1/3*k] / Gamma[2 + 1/3*k] / Exp[1] / k!, {k, 0, Infinity}] (* _Robert G. Wilson v_, Jun 13 2002 *)
%Y A072020 Cf. A072019.
%K A072020 nonn
%O A072020 1,2
%A A072020 _Karol A. Penson_, Jun 05 2002
%E A072020 a(9) from _Robert G. Wilson v_, Jun 13 2002
%E A072020 a(10) from _Sean A. Irvine_, Aug 26 2024