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 A143476 Denominator of the coefficient of z^(2n) in the Stirling-like asymptotic expansion of the hyperfactorial function A002109. 10
 1, 720, 7257600, 15676416000, 3476402012160000, 162695614169088000000, 4919915372473221120000000, 60219764159072226508800000000, 507464726196802564122476544000000000, 3288371425755280615513648005120000000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 REFERENCES 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. 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) LINKS Seiichi Manyama, Table of n, a(n) for n = 0..148 Eric Weisstein's World of Mathematics, Hyperfactorial FORMULA From Seiichi Manyama, Aug 31 2018: (Start) Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence 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. a(n) is the denominator of c_n. (End) EXAMPLE (Glaisher*(1 - 1433/(7257600*z^4) + 1/(720*z^2))*z^(1/12 + (z*(1 + z))/2))/e^(z^2/4). From Seiichi Manyama, Aug 31 2018: (Start) c_1 = -1/2 * (B_4*c_0/(3*4)) = 1/720, so a(1) = 720. c_2 = -1/4 * (B_6*c_0/(5*6) + B_4*c_1/(3*4)) = -1433/7257600, so a(2) = 7257600. (End) CROSSREFS Cf. A002109, A143475. Sequence in context: A283830 A075754 A318711 * A008979 A158044 A181751 Adjacent sequences:  A143473 A143474 A143475 * A143477 A143478 A143479 KEYWORD nonn,frac AUTHOR Eric W. Weisstein, Aug 19 2008 STATUS approved

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Last modified September 21 19:57 EDT 2020. Contains 337273 sequences. (Running on oeis4.)