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

1,2

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.

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.

Example

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

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 *)
Table[(3*n)! * Hypergeometric1F1[3 - 3*n, 4, -1]/6, {n, 1, 15}] (* Vaclav Kotesovec, Jan 17 2026 *)

Crossrefs

Cf. A072019.

Sequence in context: A332740 A178673 A028452 * A177826 A122269 A171666

Adjacent sequences: A072017 A072018 A072019 * A072021 A072022 A072023

Keyword

nonn

Author

Karol A. Penson, Jun 05 2002

Extensions

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

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

Status

approved