A051716 Numerators of Bernoulli twin numbers C(n).

1, -1, -1, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, -7, -3617, 3617, 43867, -43867, -174611, 174611, 854513, -854513, -236364091, 236364091, 8553103, -8553103, -23749461029, 23749461029, 8615841276005, -8615841276005, -7709321041217, 7709321041217, 2577687858367

Offset

0,11

Comments

The Bernoulli twin numbers C(n) are defined by C(0) = 1, then C(2n) = B(2n) + B(2n-1), C(2n+1) = -B(2n+1) - B(2n), where B() are the Bernoulli numbers A027641/A027642. The definition is due to Paul Curtz.

For denominators see A051717.

Negatives of numerators of column 1 of table described in A051714/A051715.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..640

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Formula

Numerators of differences of the sequence of rational numbers 0 followed by A164555/A027642. - Paul Curtz, Jan 29 2017

The e.g.f. of the rationals a(n)/A051717(n) is -(1/x + x^2/2 + x/(1 - exp(x)) + dilog(exp(-x))), (with dilog(x) = polylog(2, 1-x)). From integrating the e.g.f. of the z-sequence (exp(x) - (1+x))/(exp(x) -1)^2 for the Bernoulli polynomials of the second kind (A290317 / A290318). - Wolfdieter Lang, Aug 07 2017

Example

The C(n) sequence is 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ...

Maple

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;

Mathematica

c[0]= 1; c[n_?EvenQ]:= BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ]:= -BernoulliB[n] - BernoulliB[n-1]; Table[Numerator[c[n]], {n, 0, 34}] (* Jean-François Alcover, Dec 19 2011 *)

Prog

(PARI) a(n) = if (n==0, 1, nu = numerator(bernfrac(n)+bernfrac(n-1)); if (n%2, -nu, nu)); \\ Michel Marcus, Jan 29 2017
(Magma)
f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;
function A051716(n)
  if n eq 0 then return 1;
  elif (n mod 2) eq 0 then return Numerator(f(n));
  else return Numerator(-f(n));
  end if;
end function;
[A051716(n): n in [0..50]]; // G. C. Greubel, Apr 22 2023
(SageMath)
def f(n): return bernoulli(n)+bernoulli(n-1)
def A051716(n):
    if (n==0): return 1
    elif (n%2==0): return numerator(f(n))
    else: return numerator(-f(n))
[A051716(n) for n in range(51)] # G. C. Greubel, Apr 22 2023

Crossrefs

Cf. A000367, A027641, A027642, A051714, A051715.

Cf. A051717, A129825, A129826, A129724, A164555.

Sequence in context: A055928 A213145 A195567 * A226260 A102060 A102058

Adjacent sequences: A051713 A051714 A051715 * A051717 A051718 A051719

Keyword

sign,easy,nice,frac

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Dec 08 1999

Edited by N. J. A. Sloane, May 25 2008

Status

approved