A298609 - OEIS

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

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OFFSET

0,7

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)).

LINKS

Table of n, a(n) for n=0..77.

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.

EXAMPLE

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]

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}]]]

CROSSREFS

Cf. A109187, A290380, A298608.

Sequence in context: A242848 A071957 A329186 * A283978 A002655 A064891

Adjacent sequences: A298606 A298607 A298608 * A298610 A298611 A298612

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jan 23 2018

STATUS

approved