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A192366 Denominators of a companion to the Bernoulli numbers. 3

%I #13 Aug 08 2012 11:36:56

%S 1,2,2,3,6,15,30,35,70,105,210,1155,2310,5005,10010,15015,30030,

%T 255255,510510,1616615,3233230,969969,1939938,22309287,44618574,

%U 37182145,74364290,111546435,223092870,3234846615,6469693230

%N Denominators of a companion to the Bernoulli numbers.

%C For the numerators of the companion to the Bernoulli numbers and detailed information see A191754.

%F a(2*n+2)/a(2*n+1) = 2 for n>1.

%e The first rows of BC(n,m) matrix are

%e 0, 1/2, 1/2, 1/3, 1/6, 1/15,

%e 1/2, 0, -1/6, -1/6, -1/10, -1/30,

%e -1/2, -1/6, 0, 1/15, 1/15, 1/35,

%e 1/3, 1/6, 1/15, 0, -4/105, -4/105,

%e -1/6, -1/10, -1/15, -4/105, 0, 4/105,

%e 1/15, 1/30, 1/35, 4/105, 4/105, 0.

%p 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]

%t 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-Fran├žois Alcover_, Aug 08 2012 *)

%Y Cf. A191754 (numerator).

%K nonn,frac

%O 0,2

%A _Paul Curtz_, Jul 01 2011

%E Edited and Maple program added by Johannes W. Meijer, Jul 02 2011.

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