cp's OEIS Frontend

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

%I A288873 #28 Jun 26 2025 16:16:17
%S A288873 1,-2,8,0,-128,0,2048,0,-32768,0,2621440,0,-5796528128,0,939524096,0,
%T A288873 -7767448354816,0,1507258642989056,0,-95993412418797568,0,
%U A288873 7516375836686024704,0,-33265288504730187726848,0,19259875741830735724544,0,-855664510723636131971203072,0,4966694343692730467779807805440
%N A288873 Numerators of scaled Bernoulli numbers 4^n*B(n), with B(n) = A027641(n)/A027642(n).
%C A288873 The denominators seem to be given in A141459.
%C A288873 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).
%H A288873 Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%F A288873 a(n) = numerator(r(n)), with the rationals r(n) = 4^n*A027641(n)/A027642(n), n >= 0.
%F A288873 E.g.f. of {r(n)}_{n>=0}: 4*x/(exp(4*x) - 1).
%e A288873 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, ...
%p A288873 seq(numer(4^n*bernoulli(n)),n=0..28); # _Peter Luschny_, Jul 17 2017
%t A288873 Table[4^n BernoulliB[n] // Numerator, {n, 0, 30}] (* _Jean-François Alcover_, Jul 14 2018 *)
%o A288873 (PARI) a(n) = numerator(4^n*bernfrac(n)); \\ _Michel Marcus_, Jul 06 2017
%o A288873 (Python)
%o A288873 from sympy import bernoulli
%o A288873 def a(n): return -2 if n == 1 else (4**n * bernoulli(n)).numerator
%o A288873 [a(n) for n in range(31)]  # _Indranil Ghosh_, Jul 06 2017
%Y A288873 Cf. A141459, A027641/A027642, (-1)^n*A239275(n)/A141459(n) (B(2;n)), A285863/A285068 (B(3;n)).
%K A288873 sign,frac,easy
%O A288873 0,2
%A A288873 _Wolfdieter Lang_, Jul 05 2017