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A342318
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a(n) = numerator(((i^n * PolyLog(1 - n, -i) + (-i)^n * PolyLog(1 - n, i))) / (4^n - 2^n)) if n > 0 and a(0) = 1. Here i denotes the imaginary unit.
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2
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1, 1, 1, 1, 1, 5, 1, 61, 1, 1385, 1, 50521, 691, 2702765, 1, 199360981, 3617, 19391512145, 43867, 2404879675441, 174611, 370371188237525, 77683, 69348874393137901, 236364091, 15514534163557086905, 657931, 4087072509293123892361, 3392780147, 1252259641403629865468285
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OFFSET
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0,6
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COMMENTS
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The defining formula simultaneously represents the numerators of the unsigned divided Bernoulli numbers and the unsigned Euler secant numbers. Some authors consider the divided Bernoulli numbers B(n)/n to be more fundamental than B(n). For instance, B(n)/n is a p-integer for all primes p for which p - 1 does not divide n (see Ireland and Rosen).
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REFERENCES
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K. Ireland and M. Rosen, A classical introduction to modern number theory, vol. 84, Graduate Texts in Mathematics. Springer-Verlag, 2nd edition, 1990. [Prop. 15.2.4, p. 238]
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LINKS
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FORMULA
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EXAMPLE
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r(n) = 1, 1/2, 1/12, 1/56, 1/120, 5/992, 1/252, 61/16256, 1/240, 1385/261632, 1/132, ...
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MAPLE
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a := n -> `if`(n <= 2, 1, `if`(n::even, numer(abs(bernoulli(n))/n), abs(euler(n - 1)))); seq(a(n), n = 0..29);
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MATHEMATICA
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r[s_] := If[s == 0, 1, (I^s PolyLog[1 - s, -I] + (-I)^s PolyLog[1 - s, I]) / (4^s - 2^s)]; Table[r[n], {n, 0, 29}] // Numerator
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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