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

%I A298609 #14 Jan 25 2018 03:36:12
%S A298609 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,
%T A298609 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,
%U A298609 1134,0,3780,0,2268,0,324,0,9,0,420,0,6300,0,10500,0,4200,0,450,0,10,0
%N A298609 Polynomials related to the Motzkin sums for Coxeter type D, T(n, k) for n >= 0 and 0 <= k <= n.
%C A298609 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)).
%F A298609 A298608(n,k) = A109187(n,k) + T(n,k).
%F A298609 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.
%F A298609 p(n, x) = Catalan(n)*(n-1)*hypergeom([1-n, -n-1], [-n+1/2], 1/2-x/4) for n >= 2.
%e A298609 The first few polynomials are:
%e A298609 p0(x) = 0;
%e A298609 p1(x) = 0;
%e A298609 p2(x) = x;
%e A298609 p3(x) = 2 + 2*x^2;
%e A298609 p4(x) = 9*x + 3*x^3;
%e A298609 p5(x) = 8 + 24*x^2 + 4*x^4;
%e A298609 p6(x) = 50*x + 50*x^3 + 5*x^5;
%e A298609 p7(x) = 30 + 180*x^2 + 90*x^4 + 6*x^6;
%e A298609 p8(x) = 245*x + 490*x^3 + 147*x^5 + 7*x^7;
%e A298609 p9(x) = 112 + 1120*x^2 + 1120*x^4 + 224*x^6 + 8*x^8;
%e A298609 The triangle of coefficients extended by the main diagonal with zeros starts:
%e A298609 [0][  0]
%e A298609 [1][  0,    0]
%e A298609 [2][  0,    1,    0]
%e A298609 [3][  2,    0,    2,    0]
%e A298609 [4][  0,    9,    0,    3,    0]
%e A298609 [5][  8,    0,   24,    0,    4,    0]
%e A298609 [6][  0,   50,    0,   50,    0,    5,   0]
%e A298609 [7][ 30,    0,  180,    0,   90,    0,   6,  0]
%e A298609 [8][  0,  245,    0,  490,    0,  147,   0,  7,  0]
%e A298609 [9][112,    0, 1120,    0, 1120,    0, 224,  0,  8,  0]
%p A298609 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)):
%p A298609 A298609Row := n -> if n=0 then 0 elif n=1 then 0,0 else op(PolynomialTools:-CoefficientList(simplify(A298609Poly(n)), x)),0 fi:
%p A298609 seq(A298609Row(n), n=0..11);
%t A298609 P298609[n_] := If[n <= 1, 0, GegenbauerC[n - 1, -n, -x/2] (n - 1)/n];
%t A298609 Flatten[ Join[ {{0}, {0, 0}},
%t A298609   Table[ Join[ CoefficientList[ P298609[n], x], {0}], {n, 2, 10}]]]
%Y A298609 Cf. A109187, A290380, A298608.
%K A298609 nonn,tabl
%O A298609 0,7
%A A298609 _Peter Luschny_, Jan 23 2018