A288873 Numerators of scaled Bernoulli numbers 4^n*B(n), with B(n) = A027641(n)/A027642(n).

1, -2, 8, 0, -128, 0, 2048, 0, -32768, 0, 2621440, 0, -5796528128, 0, 939524096, 0, -7767448354816, 0, 1507258642989056, 0, -95993412418797568, 0, 7516375836686024704, 0, -33265288504730187726848, 0, 19259875741830735724544, 0, -855664510723636131971203072, 0, 4966694343692730467779807805440

Offset: 0

Author

Wolfdieter Lang, Jul 05 2017

Keywords

• sign
• frac
• easy

Comments

The denominators seem to be given in A141459.

See A285863 for comments on B(d;n) = d^n*B(n), for n >= 0, with e.g.f. d*x/(exp(d*x) - 1).

Examples

			The rationals r(n) begin: 1, -2, 8/3, 0, -128/15, 0, 2048/21, 0, -32768/15, 0, 2621440/33, 0, -5796528128/1365, 0, 939524096/3, 0, -7767448354816/255, 0, 1507258642989056/399, 0, -95993412418797568/165, ...
		

Links

• Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

Crossrefs

Cf. A141459, A027641/A027642, (-1)^n*A239275(n)/A141459(n) (B(2;n)), A285863/A285068 (B(3;n)).

Programs

Maple

seq(numer(4^n*bernoulli(n)),n=0..28); # Peter Luschny, Jul 17 2017

Mathematica

Table[4^n BernoulliB[n] // Numerator, {n, 0, 30}] (* Jean-François Alcover, Jul 14 2018 *)

PARI

a(n) = numerator(4^n*bernfrac(n)); \\ Michel Marcus, Jul 06 2017

Python

from sympy import bernoulli
def a(n): return -2 if n == 1 else (4**n * bernoulli(n)).numerator
[a(n) for n in range(31)]  # Indranil Ghosh, Jul 06 2017

Formula

a(n) = numerator(r(n)), with the rationals r(n) = 4^n*A027641(n)/A027642(n), n >= 0.

E.g.f. of {r(n)}_{n>=0}: 4*x/(exp(4*x) - 1).