A205810 Irregular triangle read by rows: Whitney numbers c_{n,k} (n >= 0, 0 <= k <= 2n) of Lucas lattices.

1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 3, 4, 3, 3, 1, 1, 4, 6, 8, 9, 8, 6, 4, 1, 1, 5, 10, 15, 20, 21, 20, 15, 10, 5, 1, 1, 6, 15, 26, 39, 48, 52, 48, 39, 26, 15, 6, 1, 1, 7, 21, 42, 70, 98, 119, 127, 119, 98, 70, 42, 21, 7, 1

Offset: 0

Author

N. J. A. Sloane, Jan 31 2012

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  1, 1,  1;
  1, 2,  1,  2,  1;
  1, 3,  3,  4,  3,  3,   1;
  1, 4,  6,  8,  9,  8,   6,   4,   1;
  1, 5, 10, 15, 20, 21,  20,  15,  10,  5,  1;
  1, 6, 15, 26, 39, 48,  52,  48,  39, 26, 15,  6,  1;
  1, 7, 21, 42, 70, 98, 119, 127, 119, 98, 70, 42, 21, 7, 1;
  ...
		

Links

• Mark E. AlSukaiti and Nafaa Chbili, Alexander and Jones Polynomials of weaving 3-braid links and Whitney rank polynomials of Lucas lattice, arXiv:2303.11398 [math.GT], 2023.
• E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Crossrefs

Main diagonal is A051292.

Cf. A079487, A097724.

Programs

Maple

c:= (n, k)-> `if`(k=2*n, 1, n*add(1/(n-i)*binomial(n-i, n-k+i)*binomial(k-i-1, i), i=0..floor(k/2))): seq(seq(c(n, k), k=0..2*n), n=0..8);  # Leonid Bedratyuk, May 15 2018

PARI

T(n,k) = if (k==2*n, 1, n*sum(i=0, k\2, 1/(n-i)*binomial(n-i,n-k+i)*binomial(k-i-1,i)));
tabf(nn) = for (n=0, nn, for (k=0, 2*n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 16 2018

Formula

c(n, k) = n*Sum_{i = 0..floor(k/2)} 1/(n-i)*binomial(n-i, n-k+i)*binomial(k-i-1, i) for 0 <= k <= 2*n-1; c(n, 2*n) = 1. - Leonid Bedratyuk, May 15 2018

From Peter Bala, Jun 26 2025: (Start)

For n >= 1, the n-th row polynomial R(n, x) = x^n * t(n, 1 + x + 1/x), where t(n, x) = 2*Chebyshev_T(n, x/2) (AlSukaiti and Chbili, Proposition 2.1).

Conjecture: for n >= 1, t(n, x + 1) = Sum_{k = 0..n} c(n, n-k)*t(k, x) - c(n, n). Cf. A097724. (End)