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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.
LINKS
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.
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
Sequence in context: A323852 A064231 A181130 * A156052 A170923 A083523
KEYWORD
sign,frac
AUTHOR
Peter Luschny, May 04 2012
STATUS
approved

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Last modified May 18 14:52 EDT 2024. Contains 372638 sequences. (Running on oeis4.)