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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.

%I A301475 #14 Mar 24 2018 03:23:36
%S A301475 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,
%T A301475 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,
%U A301475 20,6,1,76,69,44,20,6,1,69,44,20,6,1,44,20,6,1,20,6,1,6,1,1
%N 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.
%C A301475 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.
%F A301475 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).
%F A301475 T(n,k) is the list of the coefficients of P(n,k) in ascending order.
%F A301475 Row sums are powers of 3, row lengths are the triangular numbers.
%e A301475 Triangle of polynomials starts:
%e A301475                                     1
%e A301475                                  1 + x, 1
%e A301475                           2 + 2 x + x^2, 2 + x, 1
%e A301475                4 + 5 x + 3 x^2  + x^3, 5 + 3 x^2 + x, 3 + x, 1
%e A301475 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
%e A301475 .
%e A301475 Triangle of coefficients starts:
%e A301475                                1
%e A301475                             1, 1, 1
%e A301475                         2, 2, 1, 2, 1, 1
%e A301475                   4, 5, 3, 1, 5, 3, 1, 3, 1, 1
%e A301475          9, 12, 9, 4, 1, 12, 9, 4, 1, 9, 4, 1, 4, 1, 1
%e A301475 21, 30, 25, 14, 5, 1, 30, 25, 14, 5, 1, 25, 14, 5, 1, 14, 5, 1, 5, 1, 1
%p A301475 CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
%p A301475 T := (n,k) -> binomial(n,k)*hypergeom([-k/2,1/2-k/2], [-k+n+2], 4);
%p A301475 P := (n,m) -> add(simplify(T(n,k)*x^(n-k-m)), k=0..n-m);
%p A301475 for n from 0 to 5 do seq(sort(P(n,j),x,ascending), j=0..n) od;
%p A301475 for n from 0 to 5 do seq(CoeffList(P(n,j)), j=0..n) od;
%Y A301475 Cf. A064189 (A026300) (x=0), A038622 (x=1), A126954 (x=2), A089942 (x=-1), A301477 (x=1/2, scaled).
%Y A301475 Cf. A000244 (row sums), A000217 (row length).
%K A301475 nonn,tabf
%O A301475 0,5
%A A301475 _Peter Luschny_, Mar 22 2018