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A113081
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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.
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10
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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, 5, 1, 0, 1030091, 182402, 11973, 724, 55, 6, 1, 0, 218626341, 27392682, 1030091, 33476, 1345, 78, 7, 1, 0, 118038692523, 10390564242, 218626341, 3697844, 75695, 2246
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OFFSET
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0,8
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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Table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,...
0,1,2,3,4,5,6,7,8,9,10,11,...
0,3,10,21,36,55,78,105,136,171,210,...
0,21,114,331,724,1345,2246,3479,5096,7149,...
0,331,2970,11973,33476,75695,148926,265545,440008,...
0,11973,182402,1030091,3697844,10204145,23694838,...
0,1030091,27392682,218626341,1011973796,3416461455,...
0,218626341,10390564242,118038692523,706848765844,...
0,118038692523,10210795262650,166013096151621,...
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PROG
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(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))))))
(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])
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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