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

%I A190339 #71 Aug 25 2025 11:28:25
%S A190339 2,6,15,105,105,231,15015,2145,36465,969969,4849845,10140585,10140585,
%T A190339 22287,3231615,7713865005,7713865005,90751353,218257003965,1641030105,
%U A190339 67282234305,368217318651,1841086593255
%N A190339 The denominators of the subdiagonal in the difference table of the Bernoulli numbers.
%C A190339 Apparently a(n) = A181131(n) for n>=2 (checked numerically up to n=640). - _R. J. Mathar_, Aug 25 2025
%C A190339 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 A190339 The T(n, m) are defined by A164555(n)/A027642(n) and its successive differences, see the formulas.
%C A190339 Reading the array T(n, m), see the examples, by its antidiagonals leads to A085737(n)/A085738(n).
%C A190339 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).
%C A190339 We can get the Bernoulli numbers from the T(n, n+1) in an original way, see A192456/A191302.
%C A190339 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 A190339 The sum of the antidiagonals of T(n,m) is 1 in the first antidiagonal, otherwise 0. _Paul Curtz_, Feb 03 2015
%D A190339 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 A190339 G. C. Greubel, <a href="/A190339/b190339.txt">Table of n, a(n) for n = 0..450</a>
%H A190339 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 A190339 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">Computation and asymptotics of the Bernoulli numbers</a>
%F A190339 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 A190339 T(1, m) = A051716(m+1)/A051717(m+1);
%F A190339 T(n, n) = 2*T(n, n+1).
%F A190339 T(n+1, n+1) = (-1)^(1+n)*A181130(n+1)/A181131(n+1). - _R. J. Mathar_, Jun 18 2011
%F A190339 a(n) = A141044(n)*A181131(n). - _Paul Curtz_, Apr 21 2013
%e A190339 The first few rows of the T(n, m) array (difference table of the Bernoulli numbers) are:
%e A190339 1,       1/2,     1/6,      0,     -1/30,         0,        1/42,
%e A190339 -1/2,   -1/3,    -1/6,  -1/30,      1/30,      1/42,       -1/42,
%e A190339 1/6,     1/6,    2/15,   1/15,    -1/105,     -1/21,      -1/105,
%e A190339 0,     -1/30,   -1/15, -8/105,    -4/105,     4/105,       8/105,
%e A190339 -1/30, -1/30,  -1/105,  4/105,     8/105,     4/105,   -116/1155,
%e A190339 0,      1/42,    1/21,  4/105,    -4/105,   -32/231,     -16/231,
%e A190339 1/42,   1/42,  -1/105, -8/105, -116/1155,    16/231,  6112/15015,
%p A190339 T := proc(n,m)
%p A190339     option remember;
%p A190339     if n < 0 or m < 0 then
%p A190339         0 ;
%p A190339     elif n = 0 then
%p A190339         if m = 1 then
%p A190339             -bernoulli(m) ;
%p A190339         else
%p A190339             bernoulli(m) ;
%p A190339         end if;
%p A190339     else
%p A190339         procname(n-1,m+1)-procname(n-1,m) ;
%p A190339     end if;
%p A190339 end proc:
%p A190339 A190339 := proc(n)
%p A190339     denom( T(n+1,n)) ;
%p A190339 end proc: # _R. J. Mathar_, Apr 25 2013
%t A190339 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 A190339 (Sage)
%o A190339 def A190339_list(n) :
%o A190339     T = matrix(QQ, 2*n+1)
%o A190339     for m in (0..2*n) :
%o A190339         T[0,m] = bernoulli_polynomial(1,m)
%o A190339         for k in range(m-1,-1,-1) :
%o A190339             T[m-k,k] = T[m-k-1,k+1] - T[m-k-1,k]
%o A190339     for m in (0..n-1) : print([T[m,k] for k in (0..n-1)])
%o A190339     return [denominator(T[k,k+1]) for k in (0..n-1)]
%o A190339 A190339_list(7) # Also prints the table as displayed in EXAMPLE. _Peter Luschny_, Jun 21 2012
%K A190339 nonn,frac,changed
%O A190339 0,1
%A A190339 _Paul Curtz_, May 09 2011
%E A190339 Edited and Maple program added by _Johannes W. Meijer_, Jun 29 2011, Jun 30 2011
%E A190339 New name from _Peter Luschny_, Jun 21 2012