A212196 Numerators of the Bernoulli median numbers.

1, -1, 2, -8, 8, -32, 6112, -3712, 362624, -71706112, 3341113856, -79665268736, 1090547664896, -38770843648, 106053090598912, -5507347586961932288, 136847762542978039808, -45309996254420664320, 3447910579774800362340352, -916174777198089643491328

Offset: 0

Author

Peter Luschny, May 04 2012

Keywords

• sign
• frac

Comments

The Bernoulli median numbers are the numbers in the median (central) column of the difference table of the Bernoulli numbers.

The Sage script below is based on L. Seidel's algorithm and does not make use of a library function for the Bernoulli numbers; in fact it generates the Bernoulli numbers on the fly.

A181130 is an unsigned version with offset 1. A181131 are the denominators of the Bernoulli median numbers.

Examples

			The difference table of the Bernoulli numbers, [m] the Bernoulli median numbers.
     [1]
    1/2,  -1/2
    1/6,[-1/3],  1/6
      0,  -1/6,   1/6,       0
  -1/30, -1/30,[2/15],   -1/30,    -1/30
      0,  1/30,  1/15,   -1/15,    -1/30,         0
   1/42,  1/42,-1/105,[-8/105],   -1/105,      1/42,      1/42
      0, -1/42, -1/21,  -4/105,    4/105,      1/21,      1/42,      0
  -1/30, -1/30,-1/105,   4/105,  [8/105],     4/105,    -1/105,  -1/30, -1/30
      0,  1/30,  1/15,   8/105,    4/105,    -4/105,    -8/105,  -1/15, -1/30, 0
   5/66,  5/66, 7/165,  -4/165,-116/1155, [-32/231], -116/1155, -4/165, 7/165, ..
.
Integral_{x=0..1} 1 = 1
Integral_{x=0..1} (-1)^1*x^2 = -1/3
Integral_{x=0..1} (-1)^2*(2*x^2 - x)^2 = 2/15
Integral_{x=0..1} (-1)^3*(6*x^3 - 6*x^2 + x)^2 = -8/105,
Integral_{x=0..1} (-1)^4*(24*x^4 - 36*x^3 + 14*x^2 - x)^2 = 8/105
Integral_{x=0..1} (-1)^5*(120*x^5 - 240*x^4 + 150*x^3 - 30*x^2 + x)^2 = -32/231,
...
Integral_{x=0..1} (-1)^n*(Sum_{k=0..n} Stirling2(n,k)*k!*(-x)^k)^2 = BernoulliMedian(n).
Compare A164555. - _Peter Luschny_, Aug 13 2017
		

Links

• Peter Luschny, The computation and asymptotics of the Bernoulli numbers.
• Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

Crossrefs

Cf. A164555, A181130, A181131, A085737, A085738.

Programs

Mathematica

max = 19; t[0] = Table[ BernoulliB[n], {n, 0, 2*max}]; t[n_] := Differences[t[0], n]; a[1] = -1; a[n_] := t[n][[n + 1]] // Numerator; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jun 26 2013 *)

Sage

def BernoulliMedian_list(n) :
    def T(S, a) :
        R = [a]
        for s in S :
            a -= s
            R.append(a)
        return R
    def M(A, p) :
        R = T(A,0)
        S = add(r for r in R)
        return -S / (2*p+3)
    R = [1]; A = [1/2, -1/2]
    for k in (0..n-2) :
        A = T(A, M(A,k))
        R.append(A[k+1])
        A = T(A,0)
    return R
def A212196_list(n): return [numerator(b) for b in BernoulliMedian_list(n)]

Formula

a(n) = numerator(Sum_{k=0..n} C(n,k)*Bernoulli(n+k)). - Vladimir Kruchinin, Apr 06 2015