This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A091740 #13 Aug 30 2025 02:43:11 %S A091740 1,290,71320,22097600,8928102400,4644244774400,3046988353024000, %T A091740 2470747704449024000,2431736840968314880000,2859398101389251502080000, %U A091740 3962371103307529193881600000,6394280010754055221811609600000,11892513203530676764397417267200000,25260371493666997186451230294016000000 %N A091740 Third column (k=4) sequence of array A091534 ((5,2)-Stirling2). %H A091740 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 A091740 a(n) = A091534(n, 4), n>=2. %F A091740 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). %F A091740 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!. %F A091740 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. %t A091740 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 *) %Y A091740 Cf. A091539 (second column of array A091534 divided by 10), A007661. %K A091740 nonn,easy,changed %O A091740 2,2 %A A091740 _Wolfdieter Lang_, Feb 13 2004