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 A143476 #29 Feb 16 2025 08:33:08
%S A143476 1,720,7257600,15676416000,3476402012160000,162695614169088000000,
%T A143476 4919915372473221120000000,60219764159072226508800000000,
%U A143476 507464726196802564122476544000000000,3288371425755280615513648005120000000000
%N A143476 Denominator of the coefficient of z^(2n) in the Stirling-like asymptotic expansion of the hyperfactorial function A002109.
%C A143476 In Glaisher (1878) equation (2) is "1^1.2^2.3^3 ... n^n = A n^(n^2/2 + n/2 + 1/12) e^(-n^4/4) (1 + 1/(720n^2) - 1433/(7257600n^4) + &c.)" - _Michael Somos_, Jun 24 2012
%D A143476 Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
%D A143476 J. W. L. Glaisher, On The Product 1^1.2^2.3^3 ... n^n, Messenger of Mathematics, 7 (1878), pp. 43-47, see p. 43 eq. (2)
%H A143476 Seiichi Manyama, <a href="/A143476/b143476.txt">Table of n, a(n) for n = 0..148</a>
%H A143476 Jean-Christophe Pain, <a href="https://arxiv.org/abs/2304.07629">Series representations for the logarithm of the Glaisher-Kinkelin constant</a>, arXiv:2304.07629 [math.NT], 2023.
%H A143476 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>
%F A143476 From _Seiichi Manyama_, Aug 31 2018: (Start)
%F A143476 Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
%F A143476 c_0 = 1, c_n = (-1/(2*n)) * Sum_{k=0..n-1} B_{2*n-2*k+2}*c_k/((2*n-2*k+1)*(2*n-2*k+2)) for n > 0.
%F A143476 a(n) is the denominator of c_n. (End)
%e A143476 (Glaisher*(1 - 1433/(7257600*z^4) + 1/(720*z^2))*z^(1/12 + (z*(1 + z))/2))/e^(z^2/4).
%e A143476 From _Seiichi Manyama_, Aug 31 2018: (Start)
%e A143476 c_1 = -1/2 * (B_4*c_0/(3*4)) = 1/720, so a(1) = 720.
%e A143476 c_2 = -1/4 * (B_6*c_0/(5*6) + B_4*c_1/(3*4)) = -1433/7257600, so a(2) = 7257600. (End)
%Y A143476 Cf. A002109, A143475.
%K A143476 nonn,frac
%O A143476 0,2
%A A143476 _Eric W. Weisstein_, Aug 19 2008