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Revision History for A301475

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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.
(history; published version)

#14 by Peter Luschny at Sat Mar 24 03:23:36 EDT 2018

STATUS

proposed

approved

#13 by Peter Luschny at Thu Mar 22 16:56:49 EDT 2018

STATUS

editing

proposed

#12 by Peter Luschny at Thu Mar 22 16:55:02 EDT 2018

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.

EXAMPLE

.

Example evaluations: At x = 0 gives A064189 (A026300); at x = 1 gives A038622;

at x = 2 gives A126954; at x =-1 gives A089942; for x = 1/2 see A301477.

#11 by Peter Luschny at Thu Mar 22 16:45:13 EDT 2018

EXAMPLE

at x = 2 gives A126954; at x =-1 gives A089942; for x = 1/2 see A301477.

#10 by Peter Luschny at Thu Mar 22 16:43:09 EDT 2018

EXAMPLE

Example evaluations: At x = 0 gives A064189 (A026300); at x = 1 gives A038622; at x = 2 gives A126954; at x =-1 gives A089942.

at x = 2 gives A126954; at x =-1 gives A089942.

CROSSREFS

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

#9 by Peter Luschny at Thu Mar 22 07:26:10 EDT 2018

EXAMPLE

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

#8 by Peter Luschny at Thu Mar 22 06:52:20 EDT 2018

FORMULA

The length Row sums are powers of 3, row n is A000217(n+1)lengths are the triangular numbers.

CROSSREFS

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

#7 by Peter Luschny at Thu Mar 22 06:39:54 EDT 2018

FORMULA

The length or of row n is A000217(n+1).

#6 by Peter Luschny at Thu Mar 22 06:39:28 EDT 2018

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).

#5 by Peter Luschny at Thu Mar 22 06:38:55 EDT 2018

NAME

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

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.

The length or row n is A000217(n+1).

KEYWORD

nonn,changed,tabf