A298608 Polynomials related to the Motzkin numbers for Coxeter type D, T(n, k) for n >= 0 and 0 <= k <= n.

1, 0, 1, 2, 1, 1, 2, 6, 2, 1, 6, 9, 12, 3, 1, 8, 30, 24, 20, 4, 1, 20, 50, 90, 50, 30, 5, 1, 30, 140, 180, 210, 90, 42, 6, 1, 70, 245, 560, 490, 420, 147, 56, 7, 1, 112, 630, 1120, 1680, 1120, 756, 224, 72, 8, 1

Offset

0,4

Comments

The polynomials evaluated at x = 1 give the analog of the Motzkin numbers for Coxeter type D (see A298300 (with a shift in the indexing)).

Links

Table of n, a(n) for n=0..54.

Formula

T(n,k) = A109187(n,k) + A298609(n,k).

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

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

Example

The first few polynomials are:
p0(x) =  1;
p1(x) =  0 +     x;
p2(x) =  2 +     x +     x^2;
p3(x) =  2 +   6*x +   2*x^2 +     x^3;
p4(x) =  6 +   9*x +  12*x^2 +   3*x^3 +    x^4;
p5(x) =  8 +  30*x +  24*x^2 +  20*x^3 +   4*x^4 +     x^5;
p6(x) = 20 +  50*x +  90*x^2 +  50*x^3 +  30*x^4 +   5*x^5 +    x^6;
p7(x) = 30 + 140*x + 180*x^2 + 210*x^3 +  90*x^4 +  42*x^5 +  6*x^6 +   x^7;
The triangle starts:
[0][  1]
[1][  0,   1]
[2][  2,   1,    1]
[3][  2,   6,    2,    1]
[4][  6,   9,   12,    3,    1]
[5][  8,  30,   24,   20,    4,   1]
[6][ 20,  50,   90,   50,   30,   5,   1]
[7][ 30, 140,  180,  210,   90,  42,   6,  1]
[8][ 70, 245,  560,  490,  420, 147,  56,  7, 1]
[9][112, 630, 1120, 1680, 1120, 756, 224, 72, 8, 1]

Maple

A298608Poly := n -> `if`(n=0, 1, binomial(2*n, n)*(hypergeom([-n, -n], [-n+1/2], 1/2-x/4) + ((n-1)/(n+1))*hypergeom([-n+1, -n-1], [-n+1/2], 1/2-x/4))):
A298608Row := n -> op(PolynomialTools:-CoefficientList(simplify(A298608Poly(n)), x)): seq(A298608Row(n), n=0..9);

Mathematica

p[0] := 1;
p[n_] := GegenbauerC[n, -n , -x/2] + GegenbauerC[n - 1, -n , -x/2] (n - 1) / n;
Table[CoefficientList[p[n], x], {n, 0, 9}] // Flatten

Crossrefs

Row sums are A298300(n+1) for n >= 1.

Cf. A109187, A298609.

Sequence in context: A113186 A206497 A123218 * A327815 A007736 A107042

Adjacent sequences: A298605 A298606 A298607 * A298609 A298610 A298611

Keyword

nonn,tabl

Author

Peter Luschny, Jan 23 2018

Status

approved