cp's OEIS Frontend

A091545 First column sequence of the array (7,2)-Stirling2 A091747.

%I A091545 #14 Sep 01 2025 05:24:41
%S A091545 1,42,5544,1507968,696681216,489070213632,485157651922944,
%T A091545 646229992361361408,1112808046846264344576,2405890997281623512973312,
%U A091545 6380422924790865556405223424,20366309975932442856045473169408,77025384328976498881563979526701056,340606249502734078054275917467072069632
%N A091545 First column sequence of the array (7,2)-Stirling2 A091747.
%C A091545 Also sixth column (m=5) sequence of triangle A091543.
%H A091545 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 A091545 a(n) = Product_{j=0..n-1} ((5*j+2)*(5*j+1))/2, n>=1. From eq.12 of the Blasiak et al. reference with r=7, s=2, k=1.
%F A091545 a(n) = (5^(2*n))*risefac(1/5, n)*risefac(2/5, n)/2, n>=1, with risefac(x, n) = Pochhammer(x, n).
%F A091545 a(n) = fac5(5*n-3)*fac5(5*n-4)/2, n>=1, with fac5(5*n-4)/2 = A034323(n) and fac5(5*n-3) = A008548(n) (5-factorials).
%F A091545 E.g.f.: (hypergeom([1/5, 2/5], [], 25*x)-1)/2.
%F A091545 a(n) = A091747(n, 2), n>=1.
%F A091545 D-finite with recurrence a(n) - (5*n-3)*(5*n-4)*a(n-1) = 0. - _R. J. Mathar_, Jul 27 2022
%F A091545 a(n) ~ Pi * (5/e)^(2*n) * n^(2*n-2/5) / (Gamma(1/5) * Gamma(2/5)). - _Amiram Eldar_, Sep 01 2025
%F A091545 a(n) ~ sqrt(Pi*(1 + sqrt(5))) * 5^(2*n + 1/4) * n^(2*n - 2/5) / (Gamma(1/10) * 2^(7/10) * exp(2*n)). - _Vaclav Kotesovec_, Sep 01 2025
%t A091545 a[n_] := 5^(2*n) * Pochhammer[1/5, n] * Pochhammer[2/5, n] / 2; Array[a, 15] (* _Amiram Eldar_, Sep 01 2025 *)
%Y A091545 Cf. A008548, A034323, A091543, A091747, A175380, A246745.
%K A091545 nonn,easy,changed
%O A091545 1,2
%A A091545 _Wolfdieter Lang_, Feb 13 2004