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

0,5

Links

Table of n, a(n) for n=0..66.

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.

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

Example

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

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 *)

Prog

(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 */

Crossrefs

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

Sequence in context: A085478 A129818 A123970 * A145904 A273350 A328083

Adjacent sequences: A055895 A055896 A055897 * A055899 A055900 A055901

Keyword

nonn,tabl

Author

Christian G. Bower, Jun 13 2000

Status

approved