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A072019
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Generalized Bell numbers B_{5,2}.
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5
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1, 31, 2481, 371881, 89281461, 31274052351, 15020526041221, 9461707887414161, 7560380738419084201, 7466459670646734124671, 8925493084998518977531001, 12696331763378714706289411961, 21186586117648690791837844061341, 40976310644118022811682503135528671, 90905327647146969025291153908894514381, 229256189615621846477632508681520371943201
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
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LINKS
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FORMULA
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a(n)=sum(A091534(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+3*(j-1), 2), j=1..n), k=2..infinity)/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 .. infinity} (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)).
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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