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A091539 Second column (k=3) of array A091534 ((5,2)-Stirling2) divided by 10.

%I A091539 #20 Aug 30 2025 02:43:37
%S A091539 1,104,16192,3745280,1222291840,537758144000,307503360102400,
%T A091539 221965373351321600,197530935371241472000,212553938009841139712000,
%U A091539 272115940122123843665920000,408828811133790954169303040000,712427095375430807967713198080000,1425431682224708301179257251430400000
%N A091539 Second column (k=3) of array A091534 ((5,2)-Stirling2) divided by 10.
%H A091539 Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, <a href="https://doi.org/10.1016/S0375-9601(03)00194-4">The general boson normal ordering problem</a>, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; <a href="https://arxiv.org/abs/quant-ph/0402027">arXiv preprint</a>, arXiv:quant-ph/0402027, 2004.
%F A091539 a(n) = A091534(n, 3)/10, n >= 2.
%F A091539 a(n) = Product_{j=0..n-1} (3*j + 2)*(Product_{j=0..n-1} (3*(j+1)) - 3*Product_{j=0..n-1} (3*j + 1))/(3!*10). From eq. (12) of the Blasiak et al. reference for r=5, s=2 and k=3.
%F A091539 a(n) = (3^(2*n))*risefac(2/3, n)*(n!-3*risefac(1/3, n))/(3!*10), with risefac(x, n) = Pochhammer(x, n).
%F A091539 a(n) = (fac3(3*n-1)/10)*(fac3(3*n) - 3*fac3(3*n-2))/3!, with fac3(3*n) = A032031(n) = n!*3^n, fac3(3*n-1) = A008544(n) and fac3(3*n-2) = A007559(n) (triple factorials: fac3(n) = A007661(n)).
%F A091539 E.g.f.: (hypergeom([2/3, 1], [], 9*x)-3*hypergeom([1/3, 2/3], [], 9*x)+2)/(3!*10).
%F A091539 a(n) ~ Pi * 3^(2*n) *  n^(2*n + 2/3) / (30 * Gamma(2/3) * exp(2*n)). - _Amiram Eldar_, Aug 30 2025
%t A091539 a[n_] := 3^(2*n) * Pochhammer[2/3, n] * (n! - 3 * Pochhammer[1/3, n])/(3!*10); Array[a, 20, 2] (* _Amiram Eldar_, Aug 30 2025 *)
%Y A091539 Cf. A091534, A091540.
%Y A091539 Cf. A032031, A008544, A007559, A007661.
%K A091539 nonn,easy,changed
%O A091539 2,2
%A A091539 _Wolfdieter Lang_, Feb 13 2004