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%I #10 Jun 26 2015 21:43:40
%S 1,0,1,0,1,1,0,4,2,1,0,46,13,3,1,0,1504,242,27,4,1,0,146821,13228,693,
%T 46,5,1,0,45236404,2241527,52812,1504,70,6,1,0,46002427696,1237069018,
%U 12628008,146821,2780,99,7,1,0,159443238441379,2305369985312,9924266772
%N 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 4-tournament sequences.
%C A 4-tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = p, t_i = p (mod 3) and t_{i+1} <= 4*t_i, where p>=1. This is the table of 4-tournament sequences when the starting node has label p = k for column k>=1.
%H 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 For n>=k>0: T(n, k) = Sum_{j=1..k} T(n-1, k+3*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. Also, column k of T equals column 0 of the matrix k-th power of triangle A113095, which satisfies the matrix recurrence: A113095(n, k) = [A113095^4](n-1, k-1) + [A113095^4](n-1, k) for n>k>=0.
%e Table begins:
%e 1,1,1,1,1,1,1,1,1,1,1,1,1,...
%e 0,1,2,3,4,5,6,7,8,9,10,11,...
%e 0,4,13,27,46,70,99,133,172,216,265,...
%e 0,46,242,693,1504,2780,4626,7147,10448,14634,...
%e 0,1504,13228,52812,146821,330745,648999,1154923,1910782,...
%e 0,146821,2241527,12628008,45236404,124626530,289031301,...
%e 0,45236404,1237069018,9924266772,46002427696,155367674020,...
%e 0,46002427696,2305369985312,26507035453923,159443238441379,...
%e 0,159443238441379,14874520949557933,246323730279500082,...
%o (PARI) /* Generalized Cook-Kleber Recurrence */
%o {T(n,k,q=4)=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 for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
%o (PARI) /* Matrix Power Recurrence (Paul D. Hanna) */
%o {T(n,k,q=4)=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])}
%o for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
%Y Cf. A113095, A113096 (column 1), A113098 (column 2), A113100 (column 2); Tables: A093729 (2-tournaments), A113081 (3-tournaments), A113103 (5-tournaments); diagonals: A113093, A113094.
%K nonn,tabl
%O 0,8
%A _Paul D. Hanna_, Oct 14 2005
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