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A091549 Second column (k=3) sequence of array A078740 ((3,2)-Stirling2) divided by 6.

%I A091549 #13 Sep 01 2025 05:35:05
%S A091549 1,28,960,43200,2520000,186278400,17069875200,1902071808000,
%T A091549 253487646720000,39833773056000000,7291173820170240000,
%U A091549 1538106259064094720000,370502654756909875200000,101080724272141565952000000,31008222182732149555200000000,10627137906465962295558144000000
%N A091549 Second column (k=3) sequence of array A078740 ((3,2)-Stirling2) divided by 6.
%H A091549 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 A091549 a(n) = n!*(n+1)!*(-3 + (n+2)*(n+1)/2)/(3!)^2, n>=2.
%F A091549 E.g.f.: (hypergeom([2, 3], [], x) - 3*hypergeom([1, 2], [], x) + 2)/(3!)^2.
%F A091549 a(n) = Product_{j=0..n-1} (j+2) * (-3 * Product_{j=0..n-1} (j+1) + Product_{j=0..n-1} (j+3))/(3!)^2, n>=2. From eq.12 of the Blasiak et al. reference with r=3, s=2, k=3.
%F A091549 D-finite with recurrence a(n) + (-n^2-7*n-24)*a(n-1) + 12*(n^2+4*n+6)*a(n-2) - 36*n*(n+1)*a(n-3) = 0. - _R. J. Mathar_, Jul 27 2022
%F A091549 a(n) ~ Pi * n^(2*n+4) / (36 * exp(2*n)). - _Amiram Eldar_, Sep 01 2025
%p A091549 A091549 := proc(n)
%p A091549     n!*(n+1)!*(-3 + (n+2)*(n+1)/2)/(3!)^2 ;
%p A091549 end proc:
%p A091549 seq(A091549(n),n=2..30) ; # _R. J. Mathar_, Jul 27 2022
%t A091549 a[n_] := n!*(n+1)!*((n+2)*(n+1)/2 - 3) / 36; Array[a, 16, 2] (* _Amiram Eldar_, Sep 01 2025 *)
%Y A091549 Cf. A078740.
%K A091549 nonn,easy,changed
%O A091549 2,2
%A A091549 _Wolfdieter Lang_, Feb 13 2004