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A190339 The denominators of the subdiagonal in the difference table of the Bernoulli numbers. 26

%I

%S 2,6,15,105,105,231,15015,2145,36465,969969,4849845,10140585,10140585,

%T 22287,3231615,7713865005,7713865005,90751353,218257003965,1641030105,

%U 67282234305,368217318651,1841086593255

%N The denominators of the subdiagonal in the difference table of the Bernoulli numbers.

%C 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.

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

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

%C A164555(n)/A027642(n) is an autosequence (eigensequence such that its inverse binomial transform is the sequence signed) of the second kind; The main diagonal T(n, n) is double the first upper diagonal T(n, n+1).

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

%C 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

%C The sum of the antidiagonals of T(n,m) is 1 in the first antidiagonal, else 0. _Paul Curtz_, Feb 03 2015

%D 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.

%H Kwang-Wu Chen, <a href="http://dx.doi.org/10.1016/j.jnt.2004.08.011">A summation on Bernoulli numbers</a>, Journal of Number Theory, Volume 111, Issue 2, April 2005, Pages 372-391.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">Computation and asymptotics of the Bernoulli numbers</a>

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

%F T(1, m) = A051716(m+1)/A051717(m+1)

%F T(n, n) = 2*T(n, n+1)

%F T(n+1, n+1) = (-1)^(1+n)*A181130(n+1)/A181131(n+1) - _R. J. Mathar_, Jun 18 2011]

%F a(n) = A141044(n)*A181131(n). - _Paul Curtz_, Apr 21 2013

%e The first few rows of the T(n, m) array (difference table of the Bernoulli numbers) are:

%e 1, 1/2, 1/6, 0, -1/30, 0, 1/42,

%e -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42,

%e 1/6, 1/6, 2/15, 1/15, -1/105, -1/21, -1/105,

%e 0, -1/30, -1/15, -8/105, -4/105, 4/105, 8/105,

%e -1/30, -1/30, -1/105, 4/105, 8/105, 4/105, -116/1155,

%e 0, 1/42, 1/21, 4/105, -4/105, -32/231, -16/231,

%e 1/42, 1/42, -1/105, -8/105, -116/1155, 16/231, 6112/15015,

%p T := proc(n,m)

%p option remember;

%p if n < 0 or m < 0 then

%p 0 ;

%p elif n = 0 then

%p if m = 1 then

%p -bernoulli(m) ;

%p else

%p bernoulli(m) ;

%p end if;

%p else

%p procname(n-1,m+1)-procname(n-1,m) ;

%p end if;

%p end proc:

%p A190339 := proc(n)

%p denom( T(n+1,n)) ;

%p end proc: # _R. J. Mathar_, Apr 25 2013

%t 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 *)

%o (Sage)

%o def A190339_list(n) :

%o T = matrix(QQ, 2*n+1)

%o for m in (0..2*n) :

%o T[0,m] = bernoulli_polynomial(1,m)

%o for k in range(m-1,-1,-1) :

%o T[m-k,k] = T[m-k-1,k+1] - T[m-k-1,k]

%o for m in (0..n-1) : print [T[m,k] for k in (0..n-1)]

%o return [denominator(T[k,k+1]) for k in (0..n-1)]

%o A190339_list(7) # Also prints the table as displayed in EXAMPLE. _Peter Luschny_, Jun 21 2012

%K nonn,frac

%O 0,1

%A _Paul Curtz_, May 09 2011

%E Edited and Maple program added by _Johannes W. Meijer_, Jun 29 2011, Jun 30 2011

%E New name by _Peter Luschny_, Jun 21 2012

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Last modified September 21 15:26 EDT 2017. Contains 292303 sequences.