|
%I #28 Apr 12 2019 06:54:13
%S 1,-1,2,-8,8,-32,6112,-3712,362624,-71706112,3341113856,-79665268736,
%T 1090547664896,-38770843648,106053090598912,-5507347586961932288,
%U 136847762542978039808,-45309996254420664320,3447910579774800362340352,-916174777198089643491328
%N Numerators of the Bernoulli median numbers.
%C The Bernoulli median numbers are the numbers in the median (central) column of the difference table of the Bernoulli numbers.
%C 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.
%C A181130 is an unsigned version with offset 1. A181131 are the denominators of the Bernoulli median numbers.
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">The computation and asymptotics of the Bernoulli numbers</a>.
%H Ludwig Seidel, <a href="http://publikationen.badw.de/de/003384831">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.
%F a(n) = numerator(Sum_{k=0..n} C(n,k)*Bernoulli(n+k)). - _Vladimir Kruchinin_, Apr 06 2015
%e The difference table of the Bernoulli numbers, [m] the Bernoulli median numbers.
%e [1]
%e 1/2, -1/2
%e 1/6,[-1/3], 1/6
%e 0, -1/6, 1/6, 0
%e -1/30, -1/30,[2/15], -1/30, -1/30
%e 0, 1/30, 1/15, -1/15, -1/30, 0
%e 1/42, 1/42,-1/105,[-8/105], -1/105, 1/42, 1/42
%e 0, -1/42, -1/21, -4/105, 4/105, 1/21, 1/42, 0
%e -1/30, -1/30,-1/105, 4/105, [8/105], 4/105, -1/105, -1/30, -1/30
%e 0, 1/30, 1/15, 8/105, 4/105, -4/105, -8/105, -1/15, -1/30, 0
%e 5/66, 5/66, 7/165, -4/165,-116/1155, [-32/231], -116/1155, -4/165, 7/165, ..
%e .
%e Integral_{x=0..1} 1 = 1
%e Integral_{x=0..1} (-1)^1*x^2 = -1/3
%e Integral_{x=0..1} (-1)^2*(2*x^2 - x)^2 = 2/15
%e Integral_{x=0..1} (-1)^3*(6*x^3 - 6*x^2 + x)^2 = -8/105,
%e Integral_{x=0..1} (-1)^4*(24*x^4 - 36*x^3 + 14*x^2 - x)^2 = 8/105
%e Integral_{x=0..1} (-1)^5*(120*x^5 - 240*x^4 + 150*x^3 - 30*x^2 + x)^2 = -32/231,
%e ...
%e Integral_{x=0..1} (-1)^n*(Sum_{k=0..n} Stirling2(n,k)*k!*(-x)^k)^2 = BernoulliMedian(n).
%e Compare A164555. - _Peter Luschny_, Aug 13 2017
%t 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 *)
%o (Sage)
%o def BernoulliMedian_list(n) :
%o def T(S, a) :
%o R = [a]
%o for s in S :
%o a -= s
%o R.append(a)
%o return R
%o def M(A, p) :
%o R = T(A,0)
%o S = add(r for r in R)
%o return -S / (2*p+3)
%o R = [1]; A = [1/2, -1/2]
%o for k in (0..n-2) :
%o A = T(A, M(A,k))
%o R.append(A[k+1])
%o A = T(A,0)
%o return R
%o def A212196_list(n): return [numerator(b) for b in BernoulliMedian_list(n)]
%Y Cf. A164555, A181130, A181131, A085737, A085738.
%K sign,frac
%O 0,3
%A _Peter Luschny_, May 04 2012
|
