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A192366 Denominators of a companion to the Bernoulli numbers. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..30.

FORMULA

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

EXAMPLE

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.

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-Fran├žois Alcover, Aug 08 2012 *)

CROSSREFS

Cf. A191754 (numerator).

Sequence in context: A103687 A166678 A032908 * A060631 A275487 A096100

Adjacent sequences:  A192363 A192364 A192365 * A192367 A192368 A192369

KEYWORD

nonn,frac

AUTHOR

Paul Curtz, Jul 01 2011

EXTENSIONS

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

STATUS

approved

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Last modified December 3 04:39 EST 2016. Contains 278698 sequences.