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A113113 Number of 5-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 4 and t_i = 4 (mod 4) and t_{i+1} <= 5*t_i for 1

%I A113113 #10 Mar 14 2015 10:06:14
%S A113113 1,4,56,2704,481376,337587520,978162377600,12088945462984960,
%T A113113 651451173346940188160,155573037664478034394215424,
%U A113113 166729581953452524706695313356800
%N A113113 Number of 5-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 4 and t_i = 4 (mod 4) and t_{i+1} <= 5*t_i for 1<i<n.
%C A113113 Column 0 of triangle A113112; A113112 is the matrix 4th power of triangle A113106, which satisfies the matrix recurrence: A113106(n,k) = [A113106^5](n-1,k-1) + [A113106^5](n-1,k). Also equals column 4 of square table A113103.
%H A113113 T. D. Noe, <a href="/A113113/b113113.txt">Table of n, a(n) for n=0..30</a>
%H A113113 M. Cook and M. Kleber, <a href="http://www.combinatorics.org/Volume_7/Abstracts/v7i1r44.html">Tournament sequences and Meeussen sequences</a>, Electronic J. Comb. 7 (2000), #R44.
%e A113113 The tree of 5-tournament sequences of descendents
%e A113113 of a node labeled (4) begins:
%e A113113 [4]; generation 1: 4->[8,12,16,20];
%e A113113 generation 2: 8->[12,16,20,24,28,32,36,40],
%e A113113 12->[16,20,24,28,32,36,40,44,48,52,56,60],
%e A113113 16->[20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80],
%e A113113 20->[24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,96,100];
%e A113113 Then a(n) gives the number of nodes in generation n.
%e A113113 Also, a(n+1) = sum of labels of nodes in generation n.
%o A113113 (PARI) {a(n)=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^5)[r-1,c-1])+(M^5)[r-1,c]))); return((M^4)[n+1,1])}
%Y A113113 Cf. A008934, A113077, A113078, A113079, A113085, A113089, A113096, A113098, A113100, A113107, A113109, A113111.
%K A113113 nonn
%O A113113 0,2
%A A113113 _Paul D. Hanna_, Oct 14 2005