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

0,7

Comments

n-th row has 3n+1 entries.

Links

Table of n, a(n) for n=0..81.

S. 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.

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

Example

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

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 *)

Crossrefs

Sequence in context: A024375 A025075 A175609 * A073267 A159981 A071858

Adjacent sequences: A038714 A038715 A038716 * A038718 A038719 A038720

Keyword

nonn,tabf

Author

Floor van Lamoen, May 02 2000

Status

approved