A002427 - OEIS

%I M2510 N0993 #44 Nov 29 2022 04:45:11

%S 1,1,-1,1,-3,5,-691,35,-3617,43867,-1222277,854513,-1181820455,

%T 76977927,-23749461029,8615841276005,-84802531453387,90219075042845,

%U -26315271553053477373,38089920879940267,-261082718496449122051,1520097643918070802691

%N Numerator of (2n+1) B_{2n}, where B_n are the Bernoulli numbers.

%D A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 73.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seiichi Manyama, <a href="/A002427/b002427.txt">Table of n, a(n) for n = 0..314</a> (terms 0..100 from T. D. Noe)

%H L. Euler, <a href="http://eulerarchive.maa.org/pages/E393.html">(E393) De summis serierum numeros Bernoullianos involventium</a>, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 93.

%H M. Kaneko, <a href="https://doi.org/10.3792/pjaa.71.192">A recurrence formula for the Bernoulli numbers</a>, Proc. Japan Acad., 71 A (1995), 192-193.

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.

%e (n+1)*B_n gives: 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...

%p gf := z / (1 - exp(-z)): ser := series(gf, z, 84):

%p seq(numer((n+1)!*coeff(ser, z, n)), n=0..42, 2); # _Peter Luschny_, Aug 29 2020

%t Table[Numerator[2(2n+1)BernoulliB[2n]], {n, 1, 30}]

%o (PARI) a(n) = numerator((2*n+1)*bernfrac(2*n)); \\ _Michel Marcus_, Aug 06 2017

%o (Magma) [Numerator((2*n+1)*Bernoulli(2*n)): n in [1..30]]; // _G. C. Greubel_, Jul 03 2019

%o (Sage) [numerator((2*n+1)*bernoulli(2*n)) for n in (1..30)] # _G. C. Greubel_, Jul 03 2019

%Y Denominators are in A006955.

%Y Cf. A050925/A050932, A000367/A002445.

%K sign,easy,nice,frac

%O 0,5

%A _N. J. A. Sloane_