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