%I M1562 #32 Dec 17 2021 11:26:52

%S 1,2,6,6,10,6,210,2,30,42,110,6,546,2,30,462,170,6,51870,2,330,42,46,

%T 6,6630,22,30,798,290,6,930930,2,102,966,10,66,1919190,2,30,42,76670,

%U 6,680862,2,690,38874,470,6,46410,2,330,42,106,6,1919190

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

%C Also denominators of asymptotic expansion of polygamma function psi''(z).

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 260, (6.4.13).

%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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A006955/b006955.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 260, (6.4.13).

%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>

%F Apparently a(n) = denominator(Sum_{k=0..2*n-1} (-1)^(2*n-k+1)*E1(2*n, k+1)/ binomial(2*n, k+1)), where E1(n, k) denotes the first-order Eulerian numbers A123125. - _Peter Luschny_, Feb 17 2021

%e (n+1)*B_n gives the sequence 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, 220):

%p seq(denom((n+1)!*coeff(ser, z, n)), n=0..108, 2); # _Peter Luschny_, Aug 29 2020

%t Denominator[Table[(2n+1)BernoulliB[2n],{n,0,60}]] (* _Harvey P. Dale_, Nov 03 2011 *)

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

%Y Numerators are in A002427.

%Y Cf. A050925/A050932, A000367/A002445, A006956, A123125.

%K nonn,frac,easy,nice

%O 0,2

%A _Simon Plouffe_ and _N. J. A. Sloane_