proposed
approved
proposed
approved
editing
proposed
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.
9 + 12 x + 9 x^2 + 4 x^3 + x^4, 12 + 9 x^2 + 4 x^3 2 + x, ^3, 9 + 4 x^2 + x, ^2, 4 + x, 1
The length or of row n is A000217(n+1).
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).
Triangular array of polynomials related to the Motzkin triangle and to rooted polyominoes, coefficients in ascending order, read by rows, for 0 <= k <= n.
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.
The length or row n is A000217(n+1).
nonn,changed,tabf