A324010 The sum of squares of the number of common points in all pairs of lattice paths from (0,0) to (x,y), for x >= 0, y >= 0 (the unnormalized second moment). The table is read by antidiagonals.
1, 4, 4, 9, 26, 9, 16, 92, 92, 16, 25, 240, 474, 240, 25, 36, 520, 1704, 1704, 520, 36, 49, 994, 4879, 8084, 4879, 994, 49, 64, 1736, 11928, 29560, 29560, 11928, 1736, 64, 81, 2832, 25956, 89928, 134450, 89928, 25956, 2832, 81, 100, 4380, 51648, 238440, 498140, 498140, 238440, 51648, 4380, 100
Offset: 0
Examples
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 ... ...
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.
Crossrefs
Programs
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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 *) -
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
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.