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A301475
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Triangular array of polynomials related to the Motzkin triangle and to rooted polyominoes, coefficients in ascending order, read by rows, for 0 <= k <= n.
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1
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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
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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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
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MAPLE
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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;
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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