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A212196 Numerators of the Bernoulli median numbers.

%I A212196 #28 Apr 12 2019 06:54:13
%S A212196 1,-1,2,-8,8,-32,6112,-3712,362624,-71706112,3341113856,-79665268736,
%T A212196 1090547664896,-38770843648,106053090598912,-5507347586961932288,
%U A212196 136847762542978039808,-45309996254420664320,3447910579774800362340352,-916174777198089643491328
%N A212196 Numerators of the Bernoulli median numbers.
%C A212196 The Bernoulli median numbers are the numbers in the median (central) column of the difference table of the Bernoulli numbers.
%C A212196 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 A212196 A181130 is an unsigned version with offset 1. A181131 are the denominators of the Bernoulli median numbers.
%H A212196 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">The computation and asymptotics of the Bernoulli numbers</a>.
%H A212196 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 A212196 a(n) = numerator(Sum_{k=0..n} C(n,k)*Bernoulli(n+k)). - _Vladimir Kruchinin_, Apr 06 2015
%e A212196 The difference table of the Bernoulli numbers, [m] the Bernoulli median numbers.
%e A212196      [1]
%e A212196     1/2,  -1/2
%e A212196     1/6,[-1/3],  1/6
%e A212196       0,  -1/6,   1/6,       0
%e A212196   -1/30, -1/30,[2/15],   -1/30,    -1/30
%e A212196       0,  1/30,  1/15,   -1/15,    -1/30,         0
%e A212196    1/42,  1/42,-1/105,[-8/105],   -1/105,      1/42,      1/42
%e A212196       0, -1/42, -1/21,  -4/105,    4/105,      1/21,      1/42,      0
%e A212196   -1/30, -1/30,-1/105,   4/105,  [8/105],     4/105,    -1/105,  -1/30, -1/30
%e A212196       0,  1/30,  1/15,   8/105,    4/105,    -4/105,    -8/105,  -1/15, -1/30, 0
%e A212196    5/66,  5/66, 7/165,  -4/165,-116/1155, [-32/231], -116/1155, -4/165, 7/165, ..
%e A212196 .
%e A212196 Integral_{x=0..1} 1 = 1
%e A212196 Integral_{x=0..1} (-1)^1*x^2 = -1/3
%e A212196 Integral_{x=0..1} (-1)^2*(2*x^2 - x)^2 = 2/15
%e A212196 Integral_{x=0..1} (-1)^3*(6*x^3 - 6*x^2 + x)^2 = -8/105,
%e A212196 Integral_{x=0..1} (-1)^4*(24*x^4 - 36*x^3 + 14*x^2 - x)^2 = 8/105
%e A212196 Integral_{x=0..1} (-1)^5*(120*x^5 - 240*x^4 + 150*x^3 - 30*x^2 + x)^2 = -32/231,
%e A212196 ...
%e A212196 Integral_{x=0..1} (-1)^n*(Sum_{k=0..n} Stirling2(n,k)*k!*(-x)^k)^2 = BernoulliMedian(n).
%e A212196 Compare A164555. - _Peter Luschny_, Aug 13 2017
%t A212196 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 A212196 (Sage)
%o A212196 def BernoulliMedian_list(n) :
%o A212196     def T(S, a) :
%o A212196         R = [a]
%o A212196         for s in S :
%o A212196             a -= s
%o A212196             R.append(a)
%o A212196         return R
%o A212196     def M(A, p) :
%o A212196         R = T(A,0)
%o A212196         S = add(r for r in R)
%o A212196         return -S / (2*p+3)
%o A212196     R = [1]; A = [1/2, -1/2]
%o A212196     for k in (0..n-2) :
%o A212196         A = T(A, M(A,k))
%o A212196         R.append(A[k+1])
%o A212196         A = T(A,0)
%o A212196     return R
%o A212196 def A212196_list(n): return [numerator(b) for b in BernoulliMedian_list(n)]
%Y A212196 Cf. A164555, A181130, A181131, A085737, A085738.
%K A212196 sign,frac
%O A212196 0,3
%A A212196 _Peter Luschny_, May 04 2012