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%I #15 Mar 23 2021 04:26:34
%S 1,1,1,1,1,5,1,61,1,1385,1,50521,691,2702765,1,199360981,3617,
%T 19391512145,43867,2404879675441,174611,370371188237525,77683,
%U 69348874393137901,236364091,15514534163557086905,657931,4087072509293123892361,3392780147,1252259641403629865468285
%N 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.
%C 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).
%D 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]
%H Arnold Adelberg, Shaofang Hong and Wenli Ren, <a href="https://www.ams.org/journals/proc/2008-136-01/S0002-9939-07-09025-9/">Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences</a>, Proceedings of the American Mathematical Society, Vol. 136(1), 2008, p. 61-71.
%H Bernd C. Kellner, <a href="http://arxiv.org/abs/math/0411498">The structure of Bernoulli numbers</a>, arXiv:math/0411498 [math.NT], 2004.
%F a(2*n) = |A001067(n)| for n > 0.
%F a(2*n+1) = A000364(n) = |A122045(2*n)|.
%e r(n) = 1, 1/2, 1/12, 1/56, 1/120, 5/992, 1/252, 61/16256, 1/240, 1385/261632, 1/132, ...
%p a := n -> `if`(n <= 2, 1, `if`(n::even, numer(abs(bernoulli(n))/n), abs(euler(n - 1)))); seq(a(n), n = 0..29);
%t 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
%Y Cf. A342319 (denominator), A001067, A000364, A122045.
%K nonn,frac
%O 0,6
%A _Peter Luschny_, Mar 22 2021
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