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A143475
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Numerator of the coefficient of z^(2n) in the Stirling-like asymptotic expansion of the hyperfactorial function A002109.
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10
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1, 1, -1433, 1550887, -365236274341, 31170363588856607, -2626723351027654662151, 127061942835077684151157039, -5696145248370283185291966600124423, 254326794362835881966596504823903633657, -33203124408022060010631772664020406983485604379
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OFFSET
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0,3
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REFERENCES
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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.
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LINKS
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FORMULA
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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 numerator of c_n. (End)
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EXAMPLE
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(Glaisher*(1 - 1433/(7257600*z^4) + 1/(720*z^2))*z^(1/12 + (z*(1 + z))/2))/e^(z^2/4).
c_1 = -1/2 * (B_4*c_0/(3*4)) = 1/720, so a(1) = 1.
c_2 = -1/4 * (B_6*c_0/(5*6) + B_4*c_1/(3*4)) = -1433/7257600, so a(2) = -1433. (End)
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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