A055898 Triangle: Number of directed site animals on a square lattice with n+1 total sites and k sites supported in one particular way.

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, 28, 150, 272, 209, 76, 13, 1, 1, 36, 256, 636, 696, 376, 106, 15, 1, 1, 45, 410, 1340, 1980, 1496, 615, 141, 17, 1, 1, 55, 625, 2600, 5000, 5032, 2850, 939, 181, 19, 1, 1

Offset: 0

Author

Christian G. Bower, Jun 13 2000

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1, 1;
  1, 3, 1;
  1, 6, 5, 1;
  1,10,16, 7, 1;
  ...
		

Links

• A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189 (A_sq of Section 6).
• M. Bousquet-Mélou, New enumerative results on two-dimensional directed animals
• M. Bousquet-Mélou, New enumerative results on two-dimensional directed animals, Discr. Math., 180 (1998), 73-106.

Crossrefs

Row sums give A005773. Columns 0-8: A000012, A000217, A011863(n-1), A055899-A055904. Cf. A055905, A055907.

Programs

Mathematica

nmax = 10;
A[x_, y_] = (1/2) x ((1 - (4 x/((1 + x) (1 + x - x y))))^(-1/2) - 1);
g = A[x, y] + O[x]^(nmax+3);
row[n_] := CoefficientList[Coefficient[g, x, n+2], y];
Table[row[n], {n, 0, nmax}] // Flatten (* Jean-François Alcover, Jul 24 2018 *)

Maxima

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 */

Formula

G.f.: A(x, y)=(1/2x)((1-(4x/((1+x)(1+x-xy))))^(-1/2) - 1).

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