A065551 Triangle of Faulhaber numbers (numerators) read by rows.

1, 0, 1, 0, -1, 1, 0, 1, -1, 1, 0, -3, 3, -1, 1, 0, 5, -5, 17, -2, 1, 0, -691, 691, -118, 41, -5, 1, 0, 35, -35, 359, -44, 14, -1, 1, 0, -3617, 3617, -1237, 1519, -293, 22, -7, 1, 0, 43867, -43867, 750167, -13166, 2829, -2258, 217, -4, 1, 0, -1222277, 1222277, -627073, 1540967, -198793, 689, -235, 46, -3, 1

Offset: 0

Author

Wouter Meeussen, Dec 02 2001

Keywords

• frac
• sign
• tabl

Comments

From Wolfdieter Lang, Jun 25 2011: (Start)

In the Gessel and Viennot reference f(n,k) = a(n,k)/A065553(n,k), n>=0, k>=0.

(n+1)*f(n,k) = A(n+1,n-k), with Knuth's A(m,k) =

A093556(m,k)/A093557(m,k). See the Knuth reference given in A093556, and the W. Lang link. (End)

Examples

			Triangle begins:
{1},
{0, 1},
{0, -1, 1},
{0, 1, -1, 1},
{0, -3, 3, -1, 1},
{0, 5, -5, 17, -2, 1}.
		

Links

• Ira M. Gessel and X. G. Viennot, Determinants, paths and plane partitions, 1989, p. 27, eqn 12.2

Crossrefs

Cf. A065553.

Cf. A103438.

Formula

sum(n>=0, k>=0, f(n, k)*t^k*x^(2*n+1)/(2*n+1)! ) is the expansion of (cosh(sqrt(1+4*t)*x/2)-cosh(x/2))/t/sinh(x/2).

a(n,k)=numerator(f(n,k)).