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Search: id:A038717
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| A038717 |
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Triangular array read by rows: T(n,m) = number of ways your team can score m points in n rounds of a soccer competition (loss=0 point, draw=1 point, win=3 points), for n >= 0, 0 <= m <= 3n. |
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+0
3
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| 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 0, 1, 1, 3, 3, 4, 6, 3, 3, 3, 0, 1, 1, 4, 6, 8, 13, 12, 10, 12, 6, 4, 4, 0, 1, 1, 5, 10, 15, 25, 31, 30, 35, 30, 20, 20, 10, 5, 5, 0, 1, 1, 6, 15, 26, 45, 66, 76, 90, 96, 80, 75, 60, 35, 30, 15, 6, 6, 0, 1, 1, 7, 21, 42, 77, 126, 168, 211, 252, 252, 245, 231
(list; graph; listen)
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OFFSET
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0,7
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COMMENT
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n-th row has 3n+1 entries.
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LINKS
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S. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)
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FORMULA
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T(n, m) = T(n-1, m) + T(n-1, m-1) + T(n-1, m-3).
G.f. Sum T(n, m)*z^n*w^m = 1/(1-z(1+w+w^3)). Hence m-th column is a polynomial in n of degree m given by C(n, m) + C(m-2, 1)*C(n, m-2) + C(m-4, 2)*C(n, m-4) + C(m-6, 3)*C(n, m-6) + ... E.g. column 5 is C(n, 5)+3C(n, 3). - njas, May 24 2005
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EXAMPLE
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Triangle begins:
0...1...2...3...4...5...6...7...8...9..10..11..12 points
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1
1...1...0...1
1...2...1...2...2...0...1
1...3...3...4...6...3...3...3...0...1
1...4...6...8..13..12..10..12...6...4...4...0...1
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CROSSREFS
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Adjacent sequences: A038714 A038715 A038716 this_sequence A038718 A038719 A038720
Sequence in context: A088151 A024375 A025075 this_sequence A073267 A071858 A122864
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KEYWORD
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nonn,tabf
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AUTHOR
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Floor van Lamoen (fvlamoen(AT)hotmail.com), May 02 2000
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