A091740 Third column (k=4) sequence of array A091534 ((5,2)-Stirling2).
1, 290, 71320, 22097600, 8928102400, 4644244774400, 3046988353024000, 2470747704449024000, 2431736840968314880000, 2859398101389251502080000, 3962371103307529193881600000, 6394280010754055221811609600000, 11892513203530676764397417267200000, 25260371493666997186451230294016000000
Offset: 2
Links
- Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.
Programs
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Mathematica
a[n_] := 3^(2*n) * (6 * Pochhammer[2/3, n] * Pochhammer[1/3, n] - 4 * n! * Pochhammer[2/3, n] + n! * Pochhammer[4/3, n])/4!; Array[a, 20, 2] (* Amiram Eldar, Aug 30 2025 *)
Formula
a(n) = A091534(n, 4), n>=2.
a(n) = (3^(2*n)) * (6*risefac(2/3, n) * risefac(1/3, n) - 4*n!*risefac(2/3, n) + risefac(4/3, n)*n!)/4!, with risefac(x, n) = Pochhammer(x, n).
E.g.f.: (6*hypergeom([2/3, 1/3], [], 9*x) - 4*hypergeom([1, 2/3], [], 9*x) + hypergeom([4/3, 1], [], 9*x) - 3)/4!.
a(n) = (6*fac3(3*n-2)*fac3(3*n-1)-4*fac3(3*n-1)*fac3(3*n)+fac3(3*n)*fac3(3*n+1))/4!, n>=2, with fac3(n) = A007661(n) (triple factorials). Rewritten from eq.12 of the Blasiak et al. reference for r=5, s=2, k=4.