OFFSET
0,7
COMMENTS
n-th row has 3n+1 entries.
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.
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
KEYWORD
nonn,tabf
AUTHOR
Floor van Lamoen, May 02 2000
STATUS
approved