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!).

1, 229, 207775, 472630861, 2148321709561, 17028146983530961, 214877019857456672479, 4044349155369603186936985, 108105412214943249140163409201, 3949854849387058592656207156530781, 191308664212963089686669131219301608831

Offset: 1

Author

Karol A. Penson, Jun 05 2002

Keywords

• nonn

Examples

			a(2) = 3!*LaguerreL(3, 3,-1) = 229, special value of associated Laguerre polynomial.
		

Links

• Sean A. Irvine, Table of n, a(n) for n = 1..25
• Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.

Crossrefs

Cf. A072019.

Programs

Mathematica

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 *)

Formula

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.

Extensions

a(9) from Robert G. Wilson v, Jun 13 2002

a(10) from Sean A. Irvine, Aug 26 2024