OFFSET
0,6
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.
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
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jan 31 2012
STATUS
approved