A072019 Generalized Bell numbers B_{5,2}.

1, 31, 2481, 371881, 89281461, 31274052351, 15020526041221, 9461707887414161, 7560380738419084201, 7466459670646734124671, 8925493084998518977531001, 12696331763378714706289411961, 21186586117648690791837844061341, 40976310644118022811682503135528671, 90905327647146969025291153908894514381, 229256189615621846477632508681520371943201

Offset: 1

Author

Karol A. Penson, Jun 05 2002

Keywords

• nonn

Comments

a(n), n=1,2... can be calculated as n-th moment of a positive function on a positive half-axis. This function depends on three different hypergeometric functions of type 0F4. In Maple notation: a(n)=int( x^n*( 1/216*BesselK(1/3,2/3*sqrt(x))*(36*sqrt(3)*hypergeom([],[1/3, 4/3, 5/3, 2/3],1/243*x)*GAMMA(2/3)+8*3^(1/3)*x^(1/3)*Pi*hypergeom([],[2, 4/3, 5/3, 2/3],1/243*x)+3*3^(1/6)*GAMMA(2/3)^2*x^(2/3)*hypergeom([],[2, 7/3, 4/3, 5/3],1/243*x))/Pi/GAMMA(2/3)/exp(1)/x^(1/2) ), x=0..infinity), n=1,2....

a(2)=2!*LaguerreL(2,3,-1)=31, special value of associated Laguerre polynomial.

Links

• P. Blasiak, Karol A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
• Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
• M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Crossrefs

Cf. A072020.

Programs

Mathematica

a[n_] := Sum[ 1/9*(3^n)^2 * Gamma[n + 1/3*k + 1/3] * Gamma[n + 1/3*k + 2/3] / Gamma[4/3 + 1/3*k ] / Gamma[5/3 + 1/3*k]/k!/Exp[1], {k, 0, Infinity}]
(* Second program: *)
a[n_] := Sum[Product[FactorialPower[k+3*(j-1), 2], {j, 1, n}]/k!, {k, 2, Infinity}]/E; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 16}] (* Jean-François Alcover, Sep 01 2016 *)

Formula

a(n) = Sum_{k=2..2*n} A091534(n, k) = (Sum_{k>=2} (1/k!)*Product_{j=1..n} fallfac(k+3*(j-1), 2))/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=5, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.

Sum of an infinite series: a(n) = Sum_{k>=0} (1/9 * (3^n)^2 * Gamma(n+k/3+1/3) * Gamma(n+k/3+2/3) / (Gamma(4/3+k/3) * Gamma(5/3+k/3) * k! * exp(1))).