OFFSET
0,13
COMMENTS
Absolute values of the diagonal are in A002657. Not to be confused with the poly-Bernoulli numbers.
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]
LINKS
D. Cvijovic and H. M. Srivastava, Closed form summation of the Dowker and related sums, J. Math. Phys. 48 (2007) 043507.
FORMULA
E.g.f.: [t/(exp(t)-1)]^m*exp(t*x)=sum_{n=0..infinity} B_n^m(x)*t^n/n!.
EXAMPLE
Table of fractions B(n,m) is read along antidiagonals and starts in row n=0 and column m=0:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, -1/2, -1, -3/2, -2, -5/2, -3, -7/2, ...
0, 1/6, 5/6, 2, 11/3, 35/6, 17/2, 35/3, ...
0, 0, -1/2, -9/4, -6, -25/2, -45/2, -147/4, ...
0, -1/30, 1/10, 19/10, 251/30, 24, 274/5, 1624/15, ...
0, 0, 1/6, -3/4, -9, -475/12, -120, -294, ...
0, 1/42, -5/42, -16/21, 221/42, 4315/84, 19087/84, 720, ...
0, 0, -1/6, 5/4, 11/3, -475/12, -1375/4, -36799/24, ...
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, May 13 2007, May 17 2007
STATUS
approved