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A113103 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 5-tournament sequences.

%I A113103 #10 Jun 26 2015 21:46:02
%S A113103 1,0,1,0,1,1,0,5,2,1,0,85,16,3,1,0,4985,440,33,4,1,0,1082905,43600,
%T A113103 1251,56,5,1,0,930005021,16698560,173505,2704,85,6,1,0,3306859233805,
%U A113103 26098464448,94216515,481376,4985,120,7,1,0,50220281721033905
%N A113103 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 5-tournament sequences.
%C A113103 A 5-tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = p, t_i = p (mod 4) and t_{i+1} <= 5*t_i, where p>=1. This is the table of 5-tournament sequences when the starting node has label p = k for column k>=1.
%H A113103 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 A113103 For n>=k>0: T(n, k) = Sum_{j=1..k} T(n-1, k+4*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 A113106, which satisfies the matrix recurrence: A113106(n, k) = [A113106^5](n-1, k-1) + [A113106^5](n-1, k) for n>k>=0.
%e A113103 Table begins:
%e A113103 1,1,1,1,1,1,1,1,1,1,1,1,1,...
%e A113103 0,1,2,3,4,5,6,7,8,9,10,11,...
%e A113103 0,5,16,33,56,85,120,161,208,261,320,...
%e A113103 0,85,440,1251,2704,4985,8280,12775,18656,26109,...
%e A113103 0,4985,43600,173505,481376,1082905,2122800,3774785,6241600,...
%e A113103 0,1082905,16698560,94216515,337587520,930005021,2156566656,...
%e A113103 0,930005021,26098464448,210576669921,978162377600,...
%e A113103 0,3306859233805,172513149018752,2002383115518243,...
%e A113103 0,50220281721033905,4938593053649344000,82856383278525698433,...
%o A113103 (PARI) /* Generalized Cook-Kleber Recurrence */
%o A113103 {T(n,k,q=5)=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 A113103 for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
%o A113103 (PARI) /* Matrix Power Recurrence (Paul D. Hanna) */
%o A113103 {T(n,k,q=5)=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]))); (M^k)[n+1,1]}
%o A113103 for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
%Y A113103 Cf. A113106, A113107 (column 1), A113109 (column 2), A113111 (column 3), A113113 (column 4); Tables: A093729 (2-tournaments), A113081 (3-tournaments), A113092 (4-tournaments).
%K A113103 nonn,tabl
%O A113103 0,8
%A A113103 _Paul D. Hanna_, Oct 14 2005