cp's OEIS Frontend

A091544 First column sequence of array A091746 ((6,2)-Stirling2).

%I A091544 #13 Aug 30 2025 02:43:30
%S A091544 1,30,2700,491400,150368400,69470200800,45155630520000,
%T A091544 39285398552400000,44078217175792800000,61973973349164676800000,
%U A091544 106719182107261573449600000,220908706962031457040672000000,541226332056977069749646400000000,1548989762347068373623487996800000000
%N A091544 First column sequence of array A091746 ((6,2)-Stirling2).
%C A091544 Also fifth column (m=4) sequence of triangle A091543.
%H A091544 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 A091544 a(n) = 2^(n-1)*Product_{j=0..n-1}((2*j+1)*(4*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=6, s=2, k=1.
%F A091544 a(n) = (2^(4*n-1))*risefac(1/4, n)*risefac(1/2, n), n>=1, with risefac(x, n) = Pochhammer(x, n).
%F A091544 a(n) = fac4(4*n-3)*fac4(4*n-2)/2, n>=1, with fac4(4*n-3) = A007696(n) and fac4(4*n-2)/2 = A000407(n+1) (quartic- or 4-factorials).
%F A091544 E.g.f.: (hypergeom([1/4, 1/2], [], 16*x)-1)/2.
%F A091544 a(n) = A091746(n, 2), n>=1.
%F A091544 a(n) ~ sqrt(Pi) * 2^(4*n) * n^(2*n-1/4) / (Gamma(1/4) * exp(2*n)). - _Amiram Eldar_, Aug 30 2025
%t A091544 a[n_] := 2^(4*n-1) * Pochhammer[1/4, n] * Pochhammer[1/2, n]; Array[a, 20] (* _Amiram Eldar_, Aug 30 2025 *)
%Y A091544 Cf. A091535 (third column of A091543, first column of array A091534), A000407, A007696, A091746.
%K A091544 nonn,changed
%O A091544 1,2
%A A091544 _Wolfdieter Lang_, Feb 13 2004