A192366 Denominators of a companion to the Bernoulli numbers.
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
Offset: 0
Examples
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.
Crossrefs
Cf. A191754 (numerator).
Programs
-
Maple
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]
-
Mathematica
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 *)
Formula
a(2*n+2)/a(2*n+1) = 2 for n>1.
Extensions
Edited and Maple program added by Johannes W. Meijer, Jul 02 2011.
Comments