A113078 - OEIS

%I #11 Dec 12 2021 22:54:00

%S 1,4,26,274,4721,134899,6501536,537766009,77598500096,19821981700354,

%T 9077118324755246,7531446638893873684,11423775838657143826346,

%U 31914367054676982206368909,165251261153335414813452988541

%N Number of tournament sequences: a(n) gives the number of n-th generation descendents of a node labeled (4) in the tree of tournament sequences.

%C Equals column 4 of square table A093729. Also equals column 0 of the matrix 4th power of triangle A097710, which satisfies the matrix recurrence: A097710(n,k) = [A097710^2](n-1,k-1) + [A097710^2](n-1,k) for n>k>=0.

%H M. Cook and M. Kleber, <a href="https://doi.org/10.37236/1522">Tournament sequences and Meeussen sequences</a>, Electronic J. Comb. 7 (2000), #R44.

%e The tree of tournament sequences of descendents of a node labeled (4) begins:

%e [4]; generation 1: 4->[5,6,7,8]; generation 2: 5->[6,7,8,9,10],

%e 6->[7,8,9,10,11,12], 7->[8,9,10,11,12,13,14],

%e 8->[9,10,11,12,13,14,15,16]; ...

%e Then a(n) gives the number of nodes in generation n.

%e Also, a(n+1) = sum of labels of nodes in generation n.

%o (PARI) {a(n,q=2)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^q)[r-1,c-1])+(M^q)[r-1,c]))); return((M^4)[n+1,1])}

%Y Cf. A113077, A113079, A008934, A113089, A113096, A113098, A113100, A113107, A113109, A113111, A113113.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 14 2005