A301475 Triangular array of polynomials related to the Motzkin triangle and to rooted polyominoes, coefficients in ascending order, read by rows, for 0 <= k <= n.

1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 4, 5, 3, 1, 5, 3, 1, 3, 1, 1, 9, 12, 9, 4, 1, 12, 9, 4, 1, 9, 4, 1, 4, 1, 1, 21, 30, 25, 14, 5, 1, 30, 25, 14, 5, 1, 25, 14, 5, 1, 14, 5, 1, 5, 1, 1, 51, 76, 69, 44, 20, 6, 1, 76, 69, 44, 20, 6, 1, 69, 44, 20, 6, 1, 44, 20, 6, 1, 20, 6, 1, 6, 1, 1

Offset: 0

Author

Peter Luschny, Mar 22 2018

Keywords

• nonn
• tabf

Comments

Evaluating this triangle of polynomials at different values of x leads to interesting integer triangles. For instance at x = 0 it gives the Motzkin triangle A064189 (A026300), at x = 1 it counts rooted polyominoes A038622; at x = 2 it gives A126954 and at x =-1 gives A089942; x = 1/2 and scaling gives A301477.

Examples

			Triangle of polynomials starts:
                                    1
                                 1 + x, 1
                          2 + 2 x + x^2, 2 + x, 1
               4 + 5 x + 3 x^2  + x^3, 5 + 3 x^2 + x, 3 + x, 1
9 + 12 x + 9 x^2  + 4 x^3  + x^4, 12 + 9 x + 4 x^2 + x^3, 9 + 4 x + x^2, 4 + x, 1
.
Triangle of coefficients starts:
                               1
                            1, 1, 1
                        2, 2, 1, 2, 1, 1
                  4, 5, 3, 1, 5, 3, 1, 3, 1, 1
         9, 12, 9, 4, 1, 12, 9, 4, 1, 9, 4, 1, 4, 1, 1
21, 30, 25, 14, 5, 1, 30, 25, 14, 5, 1, 25, 14, 5, 1, 14, 5, 1, 5, 1, 1
		

Crossrefs

Cf. A064189 (A026300) (x=0), A038622 (x=1), A126954 (x=2), A089942 (x=-1), A301477 (x=1/2, scaled).

Cf. A000244 (row sums), A000217 (row length).

Programs

Maple

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
T := (n,k) -> binomial(n,k)*hypergeom([-k/2,1/2-k/2], [-k+n+2], 4);
P := (n,m) -> add(simplify(T(n,k)*x^(n-k-m)), k=0..n-m);
for n from 0 to 5 do seq(sort(P(n,j),x,ascending), j=0..n) od;
for n from 0 to 5 do seq(CoeffList(P(n,j)), j=0..n) od;

Formula

P(n,k) = Sum_{j=0..n-k}binomial(n,j)*hypergeom([-j/2,1/2-j/2],[n-j+2],4)*x^(n-j-k).

T(n,k) is the list of the coefficients of P(n,k) in ascending order.

Row sums are powers of 3, row lengths are the triangular numbers.