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.
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.
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
KEYWORD
sign,frac
AUTHOR
Peter Luschny, May 04 2012
STATUS
approved