login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A191754 Numerators of a companion to the Bernoulli numbers. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..26.

FORMULA

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.

EXAMPLE

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,

MAPLE

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]

MATHEMATICA

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 *)

CROSSREFS

Sequence in context: A155769 A247390 A142719 * A165862 A077680 A041839

Adjacent sequences:  A191751 A191752 A191753 * A191755 A191756 A191757

KEYWORD

sign,frac

AUTHOR

Paul Curtz, Jun 15 2011

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 22 19:18 EDT 2014. Contains 248400 sequences.