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)).
FORMULA
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 23 2018
STATUS
approved