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%I A298608 #21 May 27 2025 06:52:39
%S A298608 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,
%T A298608 140,180,210,90,42,6,1,70,245,560,490,420,147,56,7,1,112,630,1120,
%U A298608 1680,1120,756,224,72,8,1
%N A298608 Polynomials related to the Motzkin numbers for Coxeter type D, T(n, k) for n >= 0 and 0 <= k <= n.
%C A298608 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)).
%F A298608 T(n,k) = A109187(n,k) + A298609(n,k).
%F A298608 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.
%F A298608 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.
%e A298608 The first few polynomials are:
%e A298608 p0(x) = 1;
%e A298608 p1(x) = 0 + x;
%e A298608 p2(x) = 2 + x + x^2;
%e A298608 p3(x) = 2 + 6*x + 2*x^2 + x^3;
%e A298608 p4(x) = 6 + 9*x + 12*x^2 + 3*x^3 + x^4;
%e A298608 p5(x) = 8 + 30*x + 24*x^2 + 20*x^3 + 4*x^4 + x^5;
%e A298608 p6(x) = 20 + 50*x + 90*x^2 + 50*x^3 + 30*x^4 + 5*x^5 + x^6;
%e A298608 p7(x) = 30 + 140*x + 180*x^2 + 210*x^3 + 90*x^4 + 42*x^5 + 6*x^6 + x^7;
%e A298608 The triangle starts:
%e A298608 [0][ 1]
%e A298608 [1][ 0, 1]
%e A298608 [2][ 2, 1, 1]
%e A298608 [3][ 2, 6, 2, 1]
%e A298608 [4][ 6, 9, 12, 3, 1]
%e A298608 [5][ 8, 30, 24, 20, 4, 1]
%e A298608 [6][ 20, 50, 90, 50, 30, 5, 1]
%e A298608 [7][ 30, 140, 180, 210, 90, 42, 6, 1]
%e A298608 [8][ 70, 245, 560, 490, 420, 147, 56, 7, 1]
%e A298608 [9][112, 630, 1120, 1680, 1120, 756, 224, 72, 8, 1]
%p A298608 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))):
%p A298608 A298608Row := n -> op(PolynomialTools:-CoefficientList(simplify(A298608Poly(n)), x)): seq(A298608Row(n), n=0..9);
%t A298608 p[0] := 1;
%t A298608 p[n_] := GegenbauerC[n, -n , -x/2] + GegenbauerC[n - 1, -n , -x/2] (n - 1) / n;
%t A298608 Table[CoefficientList[p[n], x], {n, 0, 9}] // Flatten
%Y A298608 Row sums are A298300(n+1) for n >= 1.
%Y A298608 Cf. A109187, A298609.
%K A298608 nonn,tabl
%O A298608 0,4
%A A298608 _Peter Luschny_, Jan 23 2018