A126853 Table read by antidiagonals: B(n,m) is the numerator of the Bernoulli polynomial of order m and degree n evaluated at x=0.
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, 1, 19, -6, 35, -3, 1, 0, 0, -5, -3, 251, -25, 17, -7, 1, 0, -1, -1, -16, -9, 24, -45, 35, -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
Offset: 0
Examples
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, ...
Links
- D. Cvijovic and H. M. Srivastava, Closed form summation of the Dowker and related sums, J. Math. Phys. 48 (2007) 043507.
- Index entries for sequences related to Bernoulli numbers.
Programs
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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 *)
Formula
E.g.f.: [t/(exp(t)-1)]^m*exp(t*x)=sum_{n=0..infinity} B_n^m(x)*t^n/n!.
Comments