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A306687 Triangular array read by rows: The sum of squares of the number of common points in all pairs of lattice paths from (0,0) to (x,y), for 0 <= y <= x (the unnormalized second moment).

%I A306687 #27 Jul 31 2019 21:09:53
%S A306687 1,4,26,9,92,474,16,240,1704,8084,25,520,4879,29560,134450,36,994,
%T A306687 11928,89928,498140,2208612,49,1736,25956,238440,1580810,8265432,
%U A306687 36024884,64,2832,51648,568128,4442768,27055808,135873360,584988840,81,4380,95733,1242648,11320595,79443000,455434875,2220096240,9470766690
%N A306687 Triangular array read by rows: The sum of squares of the number of common points in all pairs of lattice paths from (0,0) to (x,y), for 0 <= y <= x (the unnormalized second moment).
%F A306687 T(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 A306687 T(1,1) = 26, because the two lattice paths are DR and RD. (DR,DR) and (RD,RD) have three common points, (DR,RD) and (RD,DR) have two common points, and 2*3^2+2*2^2 = 26. - _Charlie Neder_, Jun 26 2019
%e A306687 The triangle begins:
%e A306687    1,
%e A306687    4,  26,
%e A306687    9,  92,  474,
%e A306687   16, 240, 1704,  8084,
%e A306687   25, 520, 4879, 29560, 134450,
%e A306687   ...
%o A306687 (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 A306687 for (n=0, 10, for (k=0, n, print1(a(n,k), ", ")); print) \\ _Michel Marcus_, Apr 08 2019
%Y A306687 Lower triangle of the square array A324010.
%K A306687 nonn,easy,tabl
%O A306687 0,2
%A A306687 _Günter Rote_, Mar 05 2019