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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.

%I A324010 #35 Jul 31 2019 21:04:58
%S A324010 1,4,4,9,26,9,16,92,92,16,25,240,474,240,25,36,520,1704,1704,520,36,
%T A324010 49,994,4879,8084,4879,994,49,64,1736,11928,29560,29560,11928,1736,64,
%U A324010 81,2832,25956,89928,134450,89928,25956,2832,81,100,4380,51648,238440,498140,498140,238440,51648,4380,100
%N 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.
%H A324010 Kevin Buchin, Kenny Chiu, Stefan Felsner, Günter Rote, André Schulz, <a href="http://arxiv.org/abs/1903.01095">The number of convex polyominoes with given height and width</a>, arXiv:1903.01095 [math.CO], 2019.
%F A324010 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.
%e A324010 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).
%e A324010 Table begins
%e A324010    1   4    9    16     25 ...
%e A324010    4  26   92   240    520 ...
%e A324010    9  92  474  1704   4879 ...
%e A324010   16 240 1704  8084  29560 ...
%e A324010   25 520 4879 29560 134450 ...
%e A324010   ...
%t A324010 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 *)
%o A324010 (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;
%o A324010 matrix(10, 10, n, k, a(n-1,k-1)) \\ _Michel Marcus_, Apr 08 2019
%Y A324010 See A306687 for the lower triangular half of the same data, read by rows.
%Y A324010 See A091044 for the unnormalized first moment (the sum of the number of common points without squaring).
%K A324010 nonn,easy,tabl
%O A324010 0,2
%A A324010 _Günter Rote_, Feb 12 2019