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A113081 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k), in the tree of 3-tournament sequences, for n>=1.

%I A113081 #8 Jun 13 2017 22:41:34
%S A113081 1,0,1,0,1,1,0,3,2,1,0,21,10,3,1,0,331,114,21,4,1,0,11973,2970,331,36,
%T A113081 5,1,0,1030091,182402,11973,724,55,6,1,0,218626341,27392682,1030091,
%U A113081 33476,1345,78,7,1,0,118038692523,10390564242,218626341,3697844,75695,2246
%N A113081 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k), in the tree of 3-tournament sequences, for n>=1.
%C A113081 A 3-tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = p, t_i = p (mod 2) and t_{i+1} <= 3*t_i, where p>=1. This is the table of 3-tournament sequences when the starting node has label p = k for column k>=1.
%H A113081 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.
%F A113081 For n>=k>0: T(n, k) = Sum_{j=1..k} T(n-1, k+2*j); else for k>n>0: T(n, k) = Sum_{j=1..n+1}(-1)^(j-1)*C(n+1, j)*T(n, k-j); with T(0, k)=1 for k>=0. Column k of T equals column 0 of the matrix k-th power of triangle A113084, which satisfies the matrix recurrence: A113084(n, k) = [A113084^3](n-1, k-1) + [A113084^3](n-1, k) for n>k>=0.
%e A113081 Table begins:
%e A113081 1,1,1,1,1,1,1,1,1,1,1,1,1,...
%e A113081 0,1,2,3,4,5,6,7,8,9,10,11,...
%e A113081 0,3,10,21,36,55,78,105,136,171,210,...
%e A113081 0,21,114,331,724,1345,2246,3479,5096,7149,...
%e A113081 0,331,2970,11973,33476,75695,148926,265545,440008,...
%e A113081 0,11973,182402,1030091,3697844,10204145,23694838,...
%e A113081 0,1030091,27392682,218626341,1011973796,3416461455,...
%e A113081 0,218626341,10390564242,118038692523,706848765844,...
%e A113081 0,118038692523,10210795262650,166013096151621,...
%o A113081 (PARI) /* Generalized Cook-Kleber Recurrence */ T(n,k,q=3)=if(n==0,1,if(n<0 || k<=0,0,if(n==1,k, if(n>=k,sum(j=1,k,T(n-1,k+(q-1)*j)), sum(j=1,n+1,(-1)^(j-1)*binomial(n+1,j)*T(n,k-j))))))
%o A113081 (PARI) /* Matrix Power Recurrence (Paul D. Hanna) */ T(n,k,q=3)=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^k)[n+1,1])
%Y A113081 Cf. A113084, A113085 (column 1), A113089 (column 2); tables: A093729 (2-tournaments), A113092 (4-tournaments), A113103 (5-tournaments).
%K A113081 nonn,tabl
%O A113081 0,8
%A A113081 _Paul D. Hanna_, Oct 14 2005