%I #21 Jan 26 2022 09:34:24
%S 1,1,1,1,3,1,1,6,5,1,1,10,16,7,1,1,15,39,31,9,1,1,21,81,101,51,11,1,1,
%T 28,150,272,209,76,13,1,1,36,256,636,696,376,106,15,1,1,45,410,1340,
%U 1980,1496,615,141,17,1,1,55,625,2600,5000,5032,2850,939,181,19,1,1
%N Triangle: Number of directed site animals on a square lattice with n+1 total sites and k sites supported in one particular way.
%H A. J. Guttmann, <a href="https://doi.org/10.1016/S0012-365X(99)00262-9">Indicators of solvability for lattice models</a>, Discrete Math., 217 (2000), 167-189 (A_sq of Section 6).
%H M. Bousquet-Mélou, <a href="http://www.labri.fr/Perso/~bousquet/Articles/Diriges/ani.ps.gz">New enumerative results on two-dimensional directed animals</a>
%H M. Bousquet-Mélou, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00109-X">New enumerative results on two-dimensional directed animals</a>, Discr. Math., 180 (1998), 73-106.
%F G.f.: A(x, y)=(1/2x)((1-(4x/((1+x)(1+x-xy))))^(-1/2) - 1).
%F T(n,m) = Sum_{k=0..n} C(n-k,m)*Sum_{i=0..n-m-k} (-1)^(n+m-i) *C(n-m-k,i) *C(k+i,k) *C(2*i+1,i+1). - _Vladimir Kruchinin_, Jan 26 2022
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 6, 5, 1;
%e 1,10,16, 7, 1;
%e ...
%t nmax = 10;
%t A[x_, y_] = (1/2) x ((1 - (4 x/((1 + x) (1 + x - x y))))^(-1/2) - 1);
%t g = A[x, y] + O[x]^(nmax+3);
%t row[n_] := CoefficientList[Coefficient[g, x, n+2], y];
%t Table[row[n], {n, 0, nmax}] // Flatten (* _Jean-François Alcover_, Jul 24 2018 *)
%o (Maxima)
%o T(n,m):=sum(binomial(n-k,m)*sum(binomial(n-m-k,i)*(-1)^(n+m-i)*binomial(k+i,k)*binomial(2*i+1,i+1),i,0,n-m-k),k,0,n); /* _Vladimir Kruchinin_, Jan 26 2022 */
%Y Row sums give A005773. Columns 0-8: A000012, A000217, A011863(n-1), A055899-A055904. Cf. A055905, A055907.
%K nonn,tabl
%O 0,5
%A _Christian G. Bower_, Jun 13 2000