This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A072019 #18 Jan 09 2025 04:22:32
%S A072019 1,31,2481,371881,89281461,31274052351,15020526041221,
%T A072019 9461707887414161,7560380738419084201,7466459670646734124671,
%U A072019 8925493084998518977531001,12696331763378714706289411961,21186586117648690791837844061341,40976310644118022811682503135528671,90905327647146969025291153908894514381,229256189615621846477632508681520371943201
%N A072019 Generalized Bell numbers B_{5,2}.
%C A072019 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....
%C A072019 a(2)=2!*LaguerreL(2,3,-1)=31, special value of associated Laguerre polynomial.
%H A072019 P. Blasiak, Karol A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%H A072019 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.
%H A072019 M. Schork, <a href="http://dx.doi.org/10.1088/0305-4470/36/16/314">On the combinatorics of normal ordering bosonic operators and deforming it</a>, J. Phys. A 36 (2003) 4651-4665.
%F A072019 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.
%F A072019 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))).
%t A072019 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}]
%t A072019 (* Second program: *)
%t A072019 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 *)
%Y A072019 Cf. A072020.
%K A072019 nonn
%O A072019 1,2
%A A072019 _Karol A. Penson_, Jun 05 2002