A006955 Denominator of (2n+1) B_{2n}, where B_n are the Bernoulli numbers. (Formerly M1562)

1, 2, 6, 6, 10, 6, 210, 2, 30, 42, 110, 6, 546, 2, 30, 462, 170, 6, 51870, 2, 330, 42, 46, 6, 6630, 22, 30, 798, 290, 6, 930930, 2, 102, 966, 10, 66, 1919190, 2, 30, 42, 76670, 6, 680862, 2, 690, 38874, 470, 6, 46410, 2, 330, 42, 106, 6, 1919190

Offset

0,2

Comments

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

References

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).

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.

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

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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).

L. Euler, (E393) De summis serierum numeros Bernoullianos involventium, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 93.

M. Kaneko, A recurrence formula for the Bernoulli numbers, Proc. Japan Acad., 71 A (1995), 192-193.

Index entries for sequences related to Bernoulli numbers.

Formula

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

Example

(n+1)*B_n gives the sequence 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...

Maple

gf := z / (1 - exp(-z)): ser := series(gf, z, 220):
seq(denom((n+1)!*coeff(ser, z, n)), n=0..108, 2); # Peter Luschny, Aug 29 2020

Mathematica

Denominator[Table[(2n+1)BernoulliB[2n], {n, 0, 60}]] (* Harvey P. Dale, Nov 03 2011 *)

Prog

(PARI) a(n) = denominator((2*n+1)*bernfrac(2*n)); \\ Michel Marcus, Aug 06 2017

Crossrefs

Numerators are in A002427.

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

Sequence in context: A353624 A021978 A141377 * A306745 A196053 A063210

Adjacent sequences: A006952 A006953 A006954 * A006956 A006957 A006958

Keyword

nonn,frac,easy,nice

Author

Simon Plouffe and N. J. A. Sloane

Status

approved