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Search: id:A093729
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| A093729 |
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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 tournament sequences. |
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10
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| 1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 7, 7, 3, 1, 0, 41, 41, 15, 4, 1, 0, 397, 397, 123, 26, 5, 1, 0, 6377, 6377, 1656, 274, 40, 6, 1, 0, 171886, 171886, 36987, 4721, 515, 57, 7, 1, 0, 7892642, 7892642, 1391106, 134899, 10810, 867, 77, 8, 1
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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Column 1 forms A008934 (number of tournament sequences). Row sums form A093730.
A tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = 1 and t_{i+1} <= 2*t_i, where integer 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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T(0, k)=1 for k>=0, T(n, 0)=0 for n>=1; else T(n, k)=T(n, k-1)-T(n-1, k)+T(n-1, 2*k-1)+T(n-1, 2*k) for k<=n; else T(n, k)=Sum_{j=1..n+1} (-1)^(j-1)*C(n+1, j)*T(n, k-j) for k>n (Cook-Kleber).
Column k of T equals column 0 of the matrix k-th power of triangle A097710, which satisfies the matrix recurrence: A097710(n, k) = [A097710^2](n-1, k-1) + [A097710^2](n-1, k) for n>k>=0.
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EXAMPLE
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Array begins:
[1,1,1,1,1,1,1,1,1,1,1,...],
[0,1,2,3,4,5,6,7,8,9,10,...],
[0,2,7,15,26,40,57,77,100,...],
[0,7,41,123,274,515,867,1351,1988,...],
[0,41,397,1656,4721,10810,21456,38507,64126,...],
[0,397,6377,36987,134899,376175,880032,1818607,3426722,...],
[0,6377,171886,1391106,6501536,...],
[0,171886,7892642,...],
[0,7892642,627340987,...],...
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PROGRAM
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(PARI) {T(n, k)=if(n<0, 0, if(n==0, 1, if(k==0, 0, if(k<=n, T(n, k-1)-T(n-1, k)+T(n-1, 2*k-1)+T(n-1, 2*k), sum(j=1, n+1, (-1)^(j-1)*binomial(n+1, j)*T(n, k-j))))))}
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CROSSREFS
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Cf. A008934, A093730.
Cf. A113080, A113081, A113092, A113103.
Adjacent sequences: A093726 A093727 A093728 this_sequence A093730 A093731 A093732
Sequence in context: A084938 A135898 A131182 this_sequence A113080 A011265 A083747
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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), Apr 14 2004; revised Oct 14 2005
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