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A227985
Numerators of the fractional triangle T(n,k) = binomial(n-1,k)*B_k for 0 <= k < n.
0
1, 0, -1, 0, -1, 1, 0, -1, 1, -1, 0, -1, 1, -1, 0, 0, -1, 1, -5, 0, 1, 0, -1, 1, -1, 0, 1, 0, 0, -1, 1, -7, 0, 7, 0, -1, 0, -1, 1, -2, 0, 7, 0, -2, 0, 0, -1, 1, -3, 0, 7, 0, -1, 0, 3, 0, -1, 1, -5, 0, 1, 0, -1, 0, 1, 0, 0, -1, 1, -11, 0, 11, 0, -11, 0, 11, 0, -5, 0, -1, 1, -1, 0, 11, 0, -22, 0, 33, 0, -5, 0
OFFSET
0,19
COMMENTS
The n-th row's sum equals the n-th Bernoulli number (with B_1 = -1/2).
Starting from B_0 = 1, the successive B n comes from the equations written with the triangle A074909
1*B_0 +2*B_1 = 0 --> B_1 = 0 -1/2
1*B_0 +3*B_1 +3*B_2 = 0 --> B_2 = 0 -1/3 +1/2
1*B_0 +4*B_1 +6*B_2 +4*B_3 = 0 --> B_3 = 0 -1/4 +1/2 -1/4,
from the terms at the left-hand side. See A159688.
Main diagonal: 1, -1/2, 1/2, -1/4, 0, 1/12, 0, -1/12, 0, 3/20, 0, -5/12, 0, 691/420,... . After the initial 1, the numerators are given by -A050925.
EXAMPLE
Triangle begins:
1,
0, -1,
0, -1, 1,
0, -1, 1, -1,
0, -1, 1, -1, 0,
0, -1, 1, -5, 0, 1,
0 -1, 1, -1, 0, 1, 0,
0, -1, 1, -7, 0, 7, 0, -1,
0, -1, 1, -2, 0, 7, 0, -2, 0, etc.
MATHEMATICA
b[0] = 1; b[1] = -1/2; row[0] = {1}; row[1] = {0, -1/2}; row[n_] := Join[{0}, List @@ (-Sum[Binomial[n+1, k]*B[k], {k, 0, n-1}]/(n+1) // Expand) /. B -> b]; b[n_] := Total[row[n]]; Table[row[n] // Numerator, {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 12 2013 *)
PROG
(PARI) t(n, k) = if (n==1, 1, if (k== -1, 0, -bernfrac(k)*binomial(n, k)/n));
tabl(nn) = {for (n = 1, nn, for (k = -1, n-2, print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Sep 07 2013
(Magma) [1] cat [Numerator(-Binomial(n, k)*Bernoulli(k)/n): k in [-1..n-2], n in [2..15]]; // Bruno Berselli, Sep 09 2013
CROSSREFS
KEYWORD
sign,frac,tabl
AUTHOR
Paul Curtz, Aug 02 2013
EXTENSIONS
More terms from Jean-François Alcover, Aug 12 2013
STATUS
approved