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%I A191754 #19 Dec 30 2015 16:14:47
%S A191754 0,1,1,1,1,1,1,1,1,-1,-1,41,41,-589,-589,8317,8317,-869807,-869807,
%T A191754 43056421,43056421,-250158593,-250158593,67632514765,67632514765,
%U A191754 -1581439548217,-1581439548217
%N A191754 Numerators of a companion to the Bernoulli numbers.
%C A191754 The companion to the Bernoulli numbers BC(0, m) = A191754(m)/A192366(m) is, just like the Bernoulli numbers T(0, m) = A164555(m)/A027642(m), see A190339 for the T(n, m), an autosequence of the second kind, i.e., its inverse binomial transform is the sequence signed.
%C A191754 In order to construct the companion array BC(n, m) we use the following rules: the main diagonal BC(n, n) = 0, the first upper diagonal BC(n, n+1) = T(n, n+1) and recurrence relation BC(n, m) = BC(n-1, m+1) - BC(n-1, m). The companion to the Bernoulli numbers appears in the first row of the BC(n, m) array, i.e., BC(0, m) = A191754(m)/A192366(m).
%C A191754 For the denominators of the companion to the Bernoulli numbers see A192366.
%F A191754 a(2*n+2)/a(2*n+1) = A000012(n)
%F A191754 BC(n, n) = 0, BC(n, n+1) = T(n, n+1) = T(n, n)/2 and BC(n, m) = BC(n-1, m+1) - BC(n-1, m); for the T(n, n+1) see A190339.
%F A191754 BC(0, m) = A191754(m)/A192366(m), i.e., the companion to the Bernoulli numbers.
%F A191754 Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A191754(k)/A192366(k). = (-1)^(n+1)*A191754(n)/ A192366(n).
%F A191754 Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A164555(k)/A027642(k). = (-1)^n*A164555(n)/A027642(n).
%F A191754 b(n) = A191754(n)/A192366(n) + A164555(n)/A027642(n) = [1, 1, 2/3, 1/3, 2/15, 1/15, 2/35, 1/35, -2/105, -1/105, ...] leads to b(2*n)/b(2*n+1) = 2 for n>1.
%e A191754 The first few rows of the BC(n,m) matrix are:
%e A191754 0, 1/2, 1/2, 1/3, 1/6, 1/15, 1/30,
%e A191754 1/2, 0, -1/6, -1/6, -1/10, -1/30, -1/210,
%e A191754 -1/2, -1/6, 0, 1/15, 1/15, 1/35, -1/105,
%e A191754 1/3, 1/6, 1/15, 0, -4/105, -4/105, 0,
%e A191754 -1/6, -1/10, -1/15, -4/105, 0, 4/105, 4/105,
%e A191754 1/15, 1/30, 1/35, 4/105, 4/105, 0, -16/231,
%e A191754 -1/30, -1/210, 1/105, 0, -4/105, -16/231, 0,
%p A191754 nmax:=26: mmax:=nmax: A164555:=proc(n): if n=1 then 1 else numer(bernoulli(n)) fi: end: A027642:=proc(n): if n=1 then 2 else denom(bernoulli(n)) fi: end: for m from 0 to 2*mmax do T(0,m):=A164555(m)/A027642(m) od: for n from 1 to nmax do for m from 0 to 2*mmax do T(n,m):= T(n-1,m+1)-T(n-1,m) od: od: for n from 0 to nmax do BC(n,n):=0: BC(n,n+1) := T(n,n+1) od: for m from 2 to 2*mmax do for n from 0 to m-2 do BC(n,m):=BC(n,m-1) + BC(n+1,m-1) od: od: for n from 0 to 2*nmax do BC(n,0):=(-1)^(n+1)*BC(0,n) od: for m from 1 to mmax do for n from 2 to 2*nmax do BC(n,m) := BC(n,m-1) + BC(n+1,m-1) od: od: for n from 0 to nmax do seq(BC(n,m),m=0..mmax) od: seq(BC(0,n),n=0..nmax): seq(numer(BC(0,n)),n=0..nmax); # _Johannes W. Meijer_, Jul 02 2011
%t A191754 max = 26; b[n_] := BernoulliB[n]; b[1]=1/2; bb = Table[b[n], {n, 0, max}]; diff = Table[ Differences[bb, n], {n, 1, Ceiling[max/2]}]; dd = Diagonal[diff]; bc[n_, n_] = 0; bc[n_, m_] /; m < n := bc[n, m] = bc[n-1, m+1] - bc[n-1, m]; bc[n_, m_] /; m == n+1 := bc[n, m] = -dd[[n+1]]; bc[n_, m_] /; m > n+1 := bc[n, m] = bc[n, m-1] + bc[n+1, m-1]; Table[bc[0, m], {m, 0, max}] // Numerator (* _Jean-François Alcover_, Aug 08 2012 *)
%Y A191754 Cf. A000012, A027642, A164555, A190339, A191754, A192366.
%K A191754 sign,frac
%O A191754 0,12
%A A191754 _Paul Curtz_, Jun 15 2011
%E A191754 Edited by _Johannes W. Meijer_, Jul 02 2011