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Search: id:A113103
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| A113103 |
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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 5-tournament sequences. |
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9
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| 1, 0, 1, 0, 1, 1, 0, 5, 2, 1, 0, 85, 16, 3, 1, 0, 4985, 440, 33, 4, 1, 0, 1082905, 43600, 1251, 56, 5, 1, 0, 930005021, 16698560, 173505, 2704, 85, 6, 1, 0, 3306859233805, 26098464448, 94216515, 481376, 4985, 120, 7, 1, 0, 50220281721033905
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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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.
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LINKS
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M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.
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FORMULA
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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.
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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,...
0,5,16,33,56,85,120,161,208,261,320,...
0,85,440,1251,2704,4985,8280,12775,18656,26109,...
0,4985,43600,173505,481376,1082905,2122800,3774785,6241600,...
0,1082905,16698560,94216515,337587520,930005021,2156566656,...
0,930005021,26098464448,210576669921,978162377600,...
0,3306859233805,172513149018752,2002383115518243,...
0,50220281721033905,4938593053649344000,82856383278525698433,...
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PROGRAM
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(PARI) /* Generalized Cook-Kleber Recurrence */ {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))))))} (PARI) /* Matrix Power Recurrence (Paul D. Hanna) */ {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]))); return((M^k)[n+1, 1])}
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CROSSREFS
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Cf. A113106, A113107 (column 1), A113109 (column 2), A113111 (column 3), A113113 (column 4); Tables: A093729 (2-tournaments), A113081 (3-tournaments), A113092 (4-tournaments).
Sequence in context: A021196 A058512 A111560 this_sequence A033325 A126690 A104714
Adjacent sequences: A113100 A113101 A113102 this_sequence A113104 A113105 A113106
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Oct 14 2005
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