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A212196 Numerators of the Bernoulli median numbers. 18
1, -1, 2, -8, 8, -32, 6112, -3712, 362624, -71706112, 3341113856, -79665268736, 1090547664896, -38770843648, 106053090598912, -5507347586961932288, 136847762542978039808, -45309996254420664320, 3447910579774800362340352, -916174777198089643491328 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..19.

Peter Luschny, The computation and asymptotics of the Bernoulli numbers.

FORMULA

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

EXAMPLE

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

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 *)

PROG

(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)]

CROSSREFS

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

Sequence in context: A227326 A064231 A181130 * A156052 A170923 A083523

Adjacent sequences:  A212193 A212194 A212195 * A212197 A212198 A212199

KEYWORD

sign,frac

AUTHOR

Peter Luschny, May 04 2012

STATUS

approved

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Last modified November 17 18:48 EST 2018. Contains 317276 sequences. (Running on oeis4.)