%I #16 Jul 21 2019 12:48:03
%S 1,0,1,0,-1,1,0,1,-1,1,0,0,5,-3,1,0,-1,-1,2,-2,1,0,0,1,-9,11,-5,1,0,1,
%T 1,19,-6,35,-3,1,0,0,-5,-3,251,-25,17,-7,1,0,-1,-1,-16,-9,24,-45,35,
%U -4,1,0,0,7,5,221,-475,274,-147,46,-9,1,0,5,3,19,11,4315,-120,1624,-56,39,-5,1,0,0,-15,-63,-199,-475,19087,-294,967,-81,145,-11,1
%N Table read by antidiagonals: B(n,m) is the numerator of the Bernoulli polynomial of order m and degree n evaluated at x=0.
%C Absolute values of the diagonal are in A002657. Not to be confused with the poly-Bernoulli numbers.
%C Let H(k) = Sum_{i=1..k+1} 1/i. Then Seq((-1)^k*T(k,k+2),k=0..) = Seq(k!*H(k),k=0..) = 1,3/2,11/3,25/2,274/5,294,.. (Cf. A160039 and A014973). [_Peter Luschny_, Apr 30 2009]
%H D. Cvijovic and H. M. Srivastava, <a href="http://dx.doi.org/10.1063/1.2712895">Closed form summation of the Dowker and related sums</a>, J. Math. Phys. 48 (2007) 043507.
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F E.g.f.: [t/(exp(t)-1)]^m*exp(t*x)=sum_{n=0..infinity} B_n^m(x)*t^n/n!.
%e Table of fractions B(n,m) is read along antidiagonals and starts in row n=0 and column m=0:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, -1/2, -1, -3/2, -2, -5/2, -3, -7/2, ...
%e 0, 1/6, 5/6, 2, 11/3, 35/6, 17/2, 35/3, ...
%e 0, 0, -1/2, -9/4, -6, -25/2, -45/2, -147/4, ...
%e 0, -1/30, 1/10, 19/10, 251/30, 24, 274/5, 1624/15, ...
%e 0, 0, 1/6, -3/4, -9, -475/12, -120, -294, ...
%e 0, 1/42, -5/42, -16/21, 221/42, 4315/84, 19087/84, 720, ...
%e 0, 0, -1/6, 5/4, 11/3, -475/12, -1375/4, -36799/24, ...
%t B[n_, m_] := n! SeriesCoefficient[(t/(E^t-1))^m E^(t x), {t, 0, n}] /. x -> 0 // Numerator; Table[B[n-m, m], {n, 0, 12}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 21 2019 *)
%K frac,sign,tabl
%O 0,13
%A _R. J. Mathar_, May 13 2007, May 17 2007