A065551 - OEIS

%I #16 Jul 04 2020 04:13:23

%S 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,

%T -5,1,0,35,-35,359,-44,14,-1,1,0,-3617,3617,-1237,1519,-293,22,-7,1,0,

%U 43867,-43867,750167,-13166,2829,-2258,217,-4,1,0,-1222277,1222277,-627073,1540967,-198793,689,-235,46,-3,1

%N Triangle of Faulhaber numbers (numerators) read by rows.

%C From _Wolfdieter Lang_, Jun 25 2011: (Start)

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

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

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

%H Ira M. Gessel and X. G. Viennot, <a href="http://people.brandeis.edu/~gessel/homepage/papers/pp.pdf">Determinants, paths and plane partitions</a>, 1989, p. 27, eqn 12.2

%F 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).

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

%e Triangle begins:

%e {1},

%e {0, 1},

%e {0, -1, 1},

%e {0, 1, -1, 1},

%e {0, -3, 3, -1, 1},

%e {0, 5, -5, 17, -2, 1}.

%Y Cf. A065553.

%Y Cf. A103438.

%K frac,sign,tabl

%O 0,12

%A _Wouter Meeussen_, Dec 02 2001