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A317615
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Numerator of the coefficient of z^(-n) of asymptotic expansions related to hyperfactorial function H(z).
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3
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1, 0, 0, 1, 0, -1, 1, 1, -1, -2591989, 1, 143285617, -576460261, -56447987, 3300470359, 3143536325702272141, -1786064113, -75094364111707432631, 2817261842693900500163, 669559278432109042489, -328057861344412119208697, -10210033960347221028684507558893, 4858439165868890899345523
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OFFSET
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0,10
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COMMENTS
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1^1*2^2*...*n^n ~ A*n^(n^2/2+n/2+1/12)*exp(-n^2/4)*(Sum_{k>=0} b(k)/n^k)^n, where A is the Glaisher-Kinkelin constant.
a(n) is the numerator of b(n).
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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/n) * Sum_{k=0..n-2} B_{n-k+1}*c_k/((n-k-1)*(n-k+1)) for n > 0.
a(n) is the numerator of c_n.
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EXAMPLE
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1^1*2^2*...*n^n ~ A*n^(n^2/2+n/2+1/12)*exp(-n^2/4)*(1 + 1/(720*n^3) - 1/(5040*n^5) + 1/(1036800*n^6) + 1/(10080*n^7) - 1/(3628800*n^8) - 2591989/(24634368000*n^9) + ... )^n.
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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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STATUS
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approved
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