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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
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
0,2
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
See A306687 for the lower triangular half of the same data, read by rows.
See A091044 for the unnormalized first moment (the sum of the number of common points without squaring).
Sequence in context: A069549 A118069 A089539 * A117416 A174943 A173317
KEYWORD
nonn,easy,tabl
AUTHOR
Günter Rote, Feb 12 2019
STATUS
approved