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

0,6

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).

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

Table of n, a(n) for n=0..29.

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.

Formula

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

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

Example

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

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

Crossrefs

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

Sequence in context: A144268 A013988 A340472 * A246006 A050970 A335955

Adjacent sequences: A342315 A342316 A342317 * A342319 A342320 A342321

Keyword

nonn,frac

Author

Peter Luschny, Mar 22 2021

Status

approved