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 A191754 Numerators of a companion to the Bernoulli numbers. 5
 0, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 41, 41, -589, -589, 8317, 8317, -869807, -869807, 43056421, 43056421, -250158593, -250158593, 67632514765, 67632514765, -1581439548217, -1581439548217 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,12 COMMENTS 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. 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). For the denominators of the companion to the Bernoulli numbers see A192366. LINKS Table of n, a(n) for n=0..26. FORMULA a(2*n+2)/a(2*n+1) = A000012(n) 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. BC(0, m) = A191754(m)/A192366(m), i.e., the companion to the Bernoulli numbers. Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A191754(k)/A192366(k). = (-1)^(n+1)*A191754(n)/ A192366(n). Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A164555(k)/A027642(k). = (-1)^n*A164555(n)/A027642(n). 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. EXAMPLE The first few rows of the BC(n,m) matrix are: 0, 1/2, 1/2, 1/3, 1/6, 1/15, 1/30, 1/2, 0, -1/6, -1/6, -1/10, -1/30, -1/210, -1/2, -1/6, 0, 1/15, 1/15, 1/35, -1/105, 1/3, 1/6, 1/15, 0, -4/105, -4/105, 0, -1/6, -1/10, -1/15, -4/105, 0, 4/105, 4/105, 1/15, 1/30, 1/35, 4/105, 4/105, 0, -16/231, -1/30, -1/210, 1/105, 0, -4/105, -16/231, 0, MAPLE 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 MATHEMATICA 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 *) CROSSREFS Cf. A000012, A027642, A164555, A190339, A191754, A192366. Sequence in context: A142719 A155884 A284043 * A165862 A339568 A077680 Adjacent sequences: A191751 A191752 A191753 * A191755 A191756 A191757 KEYWORD sign,frac AUTHOR Paul Curtz, Jun 15 2011 EXTENSIONS Edited by Johannes W. Meijer, Jul 02 2011 STATUS approved

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Last modified December 7 21:37 EST 2023. Contains 367662 sequences. (Running on oeis4.)