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A191754
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Numerators of a companion to the Bernoulli numbers.
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5
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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
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OFFSET
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0,12
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COMMENTS
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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 eigensequence 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.
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LINKS
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Table of n, a(n) for n=0..26.
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FORMULA
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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((-1)^(n-k)*binomial(n,k)*A191754(k)/A192366(k), k=0..n) = (-1)^(n+1)*A191754(n)/ A192366(n)
sum((-1)^(n-k)*binomial(n,k)*A164555(k)/A027642(k), k=0..n) = (-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.
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EXAMPLE
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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,
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MAPLE
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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]
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MATHEMATICA
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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 *)
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CROSSREFS
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Sequence in context: A217159 A155769 A142719 * A165862 A077680 A041839
Adjacent sequences: A191751 A191752 A191753 * A191755 A191756 A191757
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KEYWORD
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sign,frac
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AUTHOR
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Paul Curtz, Jun 15 2011
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EXTENSIONS
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Edited and Maple program added by Johannes W. Meijer, Jul 02 2011.
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STATUS
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approved
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