A227985 Numerators of the fractional triangle T(n,k) = binomial(n-1,k)*B_k for 0 <= k < n.

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

Author

Paul Curtz, Aug 02 2013

Keywords

• sign
• frac
• tabl

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.

Examples

			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.
		

Crossrefs

Cf. A027641/A027642, A074909.

Programs

Magma

[1] cat [Numerator(-Binomial(n,k)*Bernoulli(k)/n): k in [-1..n-2], n in [2..15]]; // Bruno Berselli, Sep 09 2013

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 *)

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

Extensions

More terms from Jean-François Alcover, Aug 12 2013