A038717 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.
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
Offset: 0
Examples
Triangle begins: 0...1...2...3...4...5...6...7...8...9..10..11..12 points -------------------------------------------------------- 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
Links
- Steven R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.
- Szymon Lukaszyk, Four Cubes, arXiv:2007.03782 [math.GM], 2020.
Programs
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Mathematica
T[_, 0] = 1; T[n_, m_] /; 0 <= m <= 3n = T[n, m] = T[n-1, m]+T[n-1, m-1]+T[n-1, m-3]; T[, ] = 0; Table[T[n, m], {n, 0, 7}, {m, 0, 3n}] // Flatten (* Jean-François Alcover, Sep 10 2018 *)
Formula
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). - N. J. A. Sloane, May 24 2005
Comments