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

0,2

Links

Table of n, a(n) for n=0..54.

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

Adjacent sequences: A324007 A324008 A324009 * A324011 A324012 A324013

Keyword

nonn,easy,tabl

Author

Günter Rote, Feb 12 2019

Status

approved