A190339 The denominators of the subdiagonal in the difference table of the Bernoulli numbers.

2, 6, 15, 105, 105, 231, 15015, 2145, 36465, 969969, 4849845, 10140585, 10140585, 22287, 3231615, 7713865005, 7713865005, 90751353, 218257003965, 1641030105, 67282234305, 368217318651, 1841086593255

Offset

0,1

Comments

The denominators of the T(n, n+1) with T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0. For the numerators of the T(n, n+1) see A191972.

The T(n, m) are defined by A164555(n)/A027642(n) and its successive differences, see the formulas.

Reading the array T(n, m), see the examples, by its antidiagonals leads to A085737(n)/A085738(n).

A164555(n)/A027642(n) is an autosequence (eigensequence whose inverse binomial transform is the sequence signed) of the second kind; the main diagonal T(n, n) is twice the first upper diagonal T(n, n+1).

We can get the Bernoulli numbers from the T(n, n+1) in an original way, see A192456/A191302.

Also the denominators of T(n, n+1) of the table defined by A085737(n)/A085738(n), the upper diagonal, called the median Bernoulli numbers by Chen. As such, Chen proved that a(n) is even only for n=0 and n=1 and that a(n) are squarefree numbers. (see Chen link). - Michel Marcus, Feb 01 2013

The sum of the antidiagonals of T(n,m) is 1 in the first antidiagonal, otherwise 0. Paul Curtz, Feb 03 2015

References

Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

Links

G. C. Greubel, Table of n, a(n) for n = 0..450

Kwang-Wu Chen, A summation on Bernoulli numbers, Journal of Number Theory, Volume 111, Issue 2, April 2005, Pages 372-391.

Peter Luschny, Computation and asymptotics of the Bernoulli numbers

Formula

T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0.

T(1, m) = A051716(m+1)/A051717(m+1);

T(n, n) = 2*T(n, n+1).

T(n+1, n+1) = (-1)^(1+n)*A181130(n+1)/A181131(n+1). - R. J. Mathar, Jun 18 2011

a(n) = A141044(n)*A181131(n). - Paul Curtz, Apr 21 2013

Example

The first few rows of the T(n, m) array (difference table of the Bernoulli numbers) are:
1,       1/2,     1/6,      0,     -1/30,         0,        1/42,
-1/2,   -1/3,    -1/6,  -1/30,      1/30,      1/42,       -1/42,
1/6,     1/6,    2/15,   1/15,    -1/105,     -1/21,      -1/105,
0,     -1/30,   -1/15, -8/105,    -4/105,     4/105,       8/105,
-1/30, -1/30,  -1/105,  4/105,     8/105,     4/105,   -116/1155,
0,      1/42,    1/21,  4/105,    -4/105,   -32/231,     -16/231,
1/42,   1/42,  -1/105, -8/105, -116/1155,    16/231,  6112/15015,

Maple

T := proc(n, m)
    option remember;
    if n < 0 or m < 0 then
        0 ;
    elif n = 0 then
        if m = 1 then
            -bernoulli(m) ;
        else
            bernoulli(m) ;
        end if;
    else
        procname(n-1, m+1)-procname(n-1, m) ;
    end if;
end proc:
A190339 := proc(n)
    denom( T(n+1, n)) ;
end proc: # R. J. Mathar, Apr 25 2013

Mathematica

nmax = 23; b[n_] := BernoulliB[n]; b[1]=1/2; bb = Table[b[n], {n, 0, 2*nmax-1}]; diff = Table[Differences[bb, n], {n, 1, nmax}]; Diagonal[diff] // Denominator (* Jean-François Alcover, Aug 08 2012 *)

Prog

(Sage)
def A190339_list(n) :
    T = matrix(QQ, 2*n+1)
    for m in (0..2*n) :
        T[0, m] = bernoulli_polynomial(1, m)
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
    for m in (0..n-1) : print([T[m, k] for k in (0..n-1)])
    return [denominator(T[k, k+1]) for k in (0..n-1)]
A190339_list(7) # Also prints the table as displayed in EXAMPLE. Peter Luschny, Jun 21 2012

Crossrefs

Sequence in context: A356223 A319336 A007709 * A078328 A038111 A356803

Adjacent sequences: A190336 A190337 A190338 * A190340 A190341 A190342

Keyword

nonn,frac

Author

Paul Curtz, May 09 2011

Extensions

Edited and Maple program added by Johannes W. Meijer, Jun 29 2011, Jun 30 2011

New name from Peter Luschny, Jun 21 2012

Status

approved