A342318 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.

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

Offset: 0

Author

Peter Luschny, Mar 22 2021

Keywords

• nonn
• frac

Comments

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).

Examples

			r(n) = 1, 1/2, 1/12, 1/56, 1/120, 5/992, 1/252, 61/16256, 1/240, 1385/261632, 1/132, ...
		

References

• 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]

Links

• Arnold Adelberg, Shaofang Hong and Wenli Ren, Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences, Proceedings of the American Mathematical Society, Vol. 136(1), 2008, p. 61-71.
• Bernd C. Kellner, The structure of Bernoulli numbers, arXiv:math/0411498 [math.NT], 2004.

Crossrefs

Cf. A342319 (denominator), A001067, A000364, A122045.

Programs

Maple

a := n -> `if`(n <= 2, 1, `if`(n::even, numer(abs(bernoulli(n))/n), abs(euler(n - 1)))); seq(a(n), n = 0..29);

Mathematica

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

Formula

a(2*n) = |A001067(n)| for n > 0.

a(2*n+1) = A000364(n) = |A122045(2*n)|.