A298609 Polynomials related to the Motzkin sums for Coxeter type D, T(n, k) for n >= 0 and 0 <= k <= n.

0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 9, 0, 3, 0, 8, 0, 24, 0, 4, 0, 0, 50, 0, 50, 0, 5, 0, 30, 0, 180, 0, 90, 0, 6, 0, 0, 245, 0, 490, 0, 147, 0, 7, 0, 112, 0, 1120, 0, 1120, 0, 224, 0, 8, 0, 0, 1134, 0, 3780, 0, 2268, 0, 324, 0, 9, 0, 420, 0, 6300, 0, 10500, 0, 4200, 0, 450, 0, 10, 0

Offset: 0

Author

Peter Luschny, Jan 23 2018

Keywords

• nonn
• tabl

Comments

The polynomials evaluated at x = 1 give the analog of the Motzkin sums for Coxeter type D (see A290380 (with a shift in the indexing)).

Examples

			The first few polynomials are:
p0(x) = 0;
p1(x) = 0;
p2(x) = x;
p3(x) = 2 + 2*x^2;
p4(x) = 9*x + 3*x^3;
p5(x) = 8 + 24*x^2 + 4*x^4;
p6(x) = 50*x + 50*x^3 + 5*x^5;
p7(x) = 30 + 180*x^2 + 90*x^4 + 6*x^6;
p8(x) = 245*x + 490*x^3 + 147*x^5 + 7*x^7;
p9(x) = 112 + 1120*x^2 + 1120*x^4 + 224*x^6 + 8*x^8;
The triangle of coefficients extended by the main diagonal with zeros starts:
[0][  0]
[1][  0,    0]
[2][  0,    1,    0]
[3][  2,    0,    2,    0]
[4][  0,    9,    0,    3,    0]
[5][  8,    0,   24,    0,    4,    0]
[6][  0,   50,    0,   50,    0,    5,   0]
[7][ 30,    0,  180,    0,   90,    0,   6,  0]
[8][  0,  245,    0,  490,    0,  147,   0,  7,  0]
[9][112,    0, 1120,    0, 1120,    0, 224,  0,  8,  0]
		

Crossrefs

Cf. A109187, A290380, A298608.

Programs

Maple

A298609Poly := n -> `if`(n<=1, 0, binomial(2*n, n)*((n-1)/(n+1))*hypergeom([1-n, -n-1], [-n+1/2], 1/2-x/4)):
A298609Row := n -> if n=0 then 0 elif n=1 then 0,0 else op(PolynomialTools:-CoefficientList(simplify(A298609Poly(n)), x)),0 fi:
seq(A298609Row(n), n=0..11);

Mathematica

P298609[n_] := If[n <= 1, 0, GegenbauerC[n - 1, -n, -x/2] (n - 1)/n];
Flatten[ Join[ {{0}, {0, 0}},
  Table[ Join[ CoefficientList[ P298609[n], x], {0}], {n, 2, 10}]]]

Formula

A298608(n,k) = A109187(n,k) + T(n,k).

The polynomials are defined by p(0, x) = p(1, x) = 0 and for n >= 2 by p(n, x) = G(n - 1, -n, -x/2)*(n - 1)/n where G(n, a, x) denotes the n-th Gegenbauer polynomial.

p(n, x) = Catalan(n)*(n-1)*hypergeom([1-n, -n-1], [-n+1/2], 1/2-x/4) for n >= 2.