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A192366
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Denominators of a companion to the Bernoulli numbers.
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3
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1, 2, 2, 3, 6, 15, 30, 35, 70, 105, 210, 1155, 2310, 5005, 10010, 15015, 30030, 255255, 510510, 1616615, 3233230, 969969, 1939938, 22309287, 44618574, 37182145, 74364290, 111546435, 223092870, 3234846615, 6469693230
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OFFSET
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0,2
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COMMENTS
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For the numerators of the companion to the Bernoulli numbers and detailed information see A191754.
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LINKS
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FORMULA
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a(2*n+2)/a(2*n+1) = 2 for n>1.
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EXAMPLE
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The first rows of BC(n,m) matrix are
0, 1/2, 1/2, 1/3, 1/6, 1/15,
1/2, 0, -1/6, -1/6, -1/10, -1/30,
-1/2, -1/6, 0, 1/15, 1/15, 1/35,
1/3, 1/6, 1/15, 0, -4/105, -4/105,
-1/6, -1/10, -1/15, -4/105, 0, 4/105,
1/15, 1/30, 1/35, 4/105, 4/105, 0.
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MAPLE
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nmax:=30: 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(denom(BC(0, n)), n=0..nmax); [Johannes W. Meijer, Jul 02 2011]
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MATHEMATICA
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max = 30; 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}] // Denominator (* Jean-François Alcover, Aug 08 2012 *)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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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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