A055899 -id:A055899 - cp's OEIS frontend

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

Christian G. Bower, Jun 13 2000

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

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.

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

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 *)

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 */

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

A247644 Triangle formed from the odd-numbered rows of A088855.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 9, 9, 9, 3, 1, 1, 4, 16, 24, 36, 24, 16, 4, 1, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 1, 6, 36, 90, 225, 300, 400, 300, 225, 90, 36, 6, 1, 1, 7, 49, 147, 441, 735, 1225, 1225, 1225, 735, 441, 147, 49, 7, 1, 1, 8, 64, 224, 784, 1568, 3136, 3920, 4900, 3920, 3136, 1568, 784, 224, 64, 8, 1

Offset: 1

N. J. A. Sloane, Sep 23 2014

The rows give the coefficients in the numerator polynomials of the o.g.f.s for the columns of triangle A055898. - Georg Fischer, Aug 16 2021

They also occur (with a factor 2*x) in the numerator polynomials of the difference A157052-A157074. - Georg Fischer, Sep 27 2021

			Triangle begins:
1,
1,1,1,
1,2,4,2,1,
1,3,9,9,9,3,1,
1,4,16,24,36,24,16,4,1,
1,5,25,50,100,100,100,50,25,5,1,
1,6,36,90,225,300,400,300,225,90,36,6,1,
1,7,49,147,441,735,1225,1225,1225,735,441,147,49,7,1,
1,8,64,224,784,1568,3136,3920,4900,3920,3136,1568,784,224,64,8,1,
...

Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.

Cf. A055899, A055900, A055901, A055902, A055903, A055904, A088855.

Cf. A088459 (even numbered rows of A088855).

row[n_] := CoefficientList[Sum[Binomial[n, k]^2 *x^(2*k), {k, 0, n}] + Sum[Binomial[n, k]*Binomial[n, k - 1]* x^(2*k - 1), {k, 0, n}], x];
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 07 2018 *)

T(n, k) = binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2); \\ A088855
row(n) = vector(2*n-1, k, T(2*n-1, k)); \\ Michel Marcus, Sep 27 2021

Row n=8 corrected by Jean-François Alcover, Jun 07 2018

Offset changed to 1 by Georg Fischer, Sep 27 2021

cp's OEIS Frontend

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

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