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A091550 Second column (k=3) sequence of array A091746 ((6,2)-Stirling2) divided by 12.

%I A091550 #13 Aug 30 2025 02:43:23
%S A091550 1,160,39900,15120000,8202070800,6058891238400,5860547004312000,
%T A091550 7196668193594880000,10944624305020966560000,
%U A091550 20199809308312018344960000,44490168120726255724917120000,115290834599202214240544256000000,347284815748143369922163257920000000
%N A091550 Second column (k=3) sequence of array A091746 ((6,2)-Stirling2) divided by 12.
%H A091550 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 A091550 a(n) = (2^(4*n)) * risefac(1/2, n) * (-3*risefac(1/4, n) + risefac(3/4, n))/(3!*12), n>=2, with risefac(x, n) = Pochhammer(x, n).
%F A091550 E.g.f.: (hypergeom([1/2, 3/4], [], 16*x) - 3*hypergeom([1/4, 1/2], [], 16*x) + 2)/(3!*12).
%F A091550 a(n) = (2^n) * Product_{j=0..n-1} (2*j+1) * (-3*Product_{j=0..n-1} (4*j+1) + Product_{j=0..n-1} (4*j+3))/(3!*12), n>=2. From eq.12 of the Blasiak et al. reference with r=6, s=2, k=3.
%F A091550 a(n) ~ sqrt(Pi) * 2^(4*n-2) * n^(2*n+1/4) / (9 * Gamma(3/4) * exp(2*n)). - _Amiram Eldar_, Aug 30 2025
%t A091550 a[n_] := 2^(4*n) * Pochhammer[1/2, n] * (-3 * Pochhammer[1/4, n] + Pochhammer[3/4, n])/(3!*12); Array[a, 20, 2] (* _Amiram Eldar_, Aug 30 2025 *)
%Y A091550 Cf. A091539 (second column of (5, 2)-Stirling2 array), A091550 (second column of (7, 2)-Stirling2 array), A091746.
%K A091550 nonn,easy,changed
%O A091550 2,2
%A A091550 _Wolfdieter Lang_, Feb 13 2004