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

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..10.

M. 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, in Maple notation: a(n)=int(x^n*(exp(-x^(1/3))*BesselI(3, 2*x^(1/6))/(3*exp(1)*x^(7/6))), x=0..infinity), n=1, 2... This representation is unique.

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 *)

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

Status

approved