OFFSET
0,2
LINKS
Kevin Buchin, Kenny Chiu, Stefan Felsner, Günter Rote, André Schulz, The number of convex polyominoes with given height and width, arXiv:1903.01095 [math.CO], 2019.
FORMULA
A(x,y) = (x+y+1) * binomial(x+y+2,x+1) * binomial(x+y,x) - binomial(2*x+2*y+2,2*x+1)/2.
EXAMPLE
There are two lattice paths from (0,0) to (x,y)=(1,1): P1=(0,0),(1,0),(1,1) and P2=(0,0),(0,1),(1,1), and hence 4 pairs of lattice paths: (P1,P1),(P1,P2),(P2,P1),(P2,P2). The number of common points is 3,2,2,3, respectively, and the sum of the squares of these numbers is 9+4+4+9 = 26 = a(1,1).
Table begins
1 4 9 16 25 ...
4 26 92 240 520 ...
9 92 474 1704 4879 ...
16 240 1704 8084 29560 ...
25 520 4879 29560 134450 ...
...
MATHEMATICA
Table[(# + y + 1) Binomial[# + y + 2, # + 1] Binomial[# + y, #] - Binomial[2 # + 2 y + 2, 2 # + 1]/2 &[x - y], {x, 0, 9}, {y, 0, x}] // Flatten (* Michael De Vlieger, Apr 15 2019 *)
PROG
(PARI) a(x, y) = (x+y+1)*binomial(x+y+2, x+1)*binomial(x+y, x)-binomial(2*x+2*y+2, 2*x+1)/2;
matrix(10, 10, n, k, a(n-1, k-1)) \\ Michel Marcus, Apr 08 2019
CROSSREFS
KEYWORD
AUTHOR
Günter Rote, Feb 12 2019
STATUS
approved