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

0,6

Links

Table of n, a(n) for n=0..63.

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.

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

Example

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;
  ...

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

Prog

(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

Crossrefs

Main diagonal is A051292.

Sequence in context: A075119 A224076 A137278 * A139368 A266506 A134303

Adjacent sequences: A205807 A205808 A205809 * A205811 A205812 A205813

Keyword

nonn,tabf

Author

N. J. A. Sloane, Jan 31 2012

Status

approved