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A113079 Number of tournament sequences: a(n) gives the number of n-th generation descendents of a node labeled (5) in the tree of tournament sequences. 11

%I #9 Dec 12 2021 22:55:05

%S 1,5,40,515,10810,376175,22099885,2231417165,393643922005,

%T 123097221805100,69087264010363930,70321483026073531730,

%U 130954011392485408662370,449450774746306949114288795

%N Number of tournament sequences: a(n) gives the number of n-th generation descendents of a node labeled (5) in the tree of tournament sequences.

%C Equals column 5 of square table A093729. Also equals column 0 of the matrix 5th 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 (5) begins:

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

%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], 9->[10,11,12,13,14,15,16,17,18],

%e 10->[11,12,13,14,15,16,17,18,19,20]; ...

%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^5)[n+1,1])}

%Y Cf. A113077, A113078, A008934, A113089, A113096, A113098, A113100, A113107, A113109, A113111, A113113.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 14 2005

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)