A290380 -id:A290380 - cp's OEIS frontend

A298300 Analog of Motzkin numbers for Coxeter type D.

1, 4, 11, 31, 87, 246, 699, 1996, 5723, 16468, 47533, 137567, 399073, 1160082, 3378483, 9855207, 28790403, 84218052, 246651729, 723165765, 2122391109, 6234634266, 18330019029, 53932825926, 158802303429, 467898288676, 1379485436579, 4069450219561

Offset: 2

F. Chapoton, Jan 16 2018

nonn

Cf. A001006 (type A), A002426 (type B), A290380.

A298300 := proc(n)
    hypergeom([(1-n)/2,1-n/2],[1],4)+(n-2)*hypergeom([1-n/2,3/2-n/2],[2],4);
    simplify(%) ;
end proc:
seq(A298300(n),n=2..40) ; # R. J. Mathar, Jul 27 2022

b[n_] := Hypergeometric2F1[(1 - n)/2, 1 - n/2, 1, 4];
c[n_] := (n-2) Hypergeometric2F1[1 - n/2, 3/2 - n/2, 2, 4];
Table[b[n] + c[n], {n, 2, 29}] (* Peter Luschny, Jan 23 2018 *)

def a(n):
     return (sum(ZZ(n - 2) / i * binomial(2 * i - 2, i - 1) *
         binomial(n - 2, 2 * i - 2)
                 for i in range(1, floor(n / 2) + 1)) +
             sum(binomial(n - 1, k) * binomial(n - 1 - k, k)
                 for k in range(floor((n - 1) / 2) + 1)))

a(n) = A002426(n-1) + A290380(n) (the latter being extended by A290380(2)=0).
Conjectural algebraic equation: 3*t+2+(3*t^2+5*t-2)*f(t)+(3*t^3-t^2)*f(t)^2 = 0.
From Peter Luschny, Jan 23 2018: (Start)
a(n) = hypergeom([(1-n)/2,1-n/2],[1],4)+(n-2)*hypergeom([1-n/2,3/2-n/2],[2],4).
a(n) = G(n-1,1-n,-1/2) + G(n-2,1-n,-1/2)*(n-2)/(n-1) where G(n,a,x) denotes the n-th Gegenbauer polynomial. (End)
D-finite with recurrence +2*n*a(n) +(-7*n+6)*a(n-1) +9*(n-4)*a(n-3)=0. - R. J. Mathar, Jul 27 2022

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

Original entry on oeis.org

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, 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, 1134, 0, 3780, 0, 2268, 0, 324, 0, 9, 0, 420, 0, 6300, 0, 10500, 0, 4200, 0, 450, 0, 10, 0

Offset: 0

Peter Luschny, Jan 23 2018

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)).

			The first few polynomials are:
p0(x) = 0;
p1(x) = 0;
p2(x) = x;
p3(x) = 2 + 2*x^2;
p4(x) = 9*x + 3*x^3;
p5(x) = 8 + 24*x^2 + 4*x^4;
p6(x) = 50*x + 50*x^3 + 5*x^5;
p7(x) = 30 + 180*x^2 + 90*x^4 + 6*x^6;
p8(x) = 245*x + 490*x^3 + 147*x^5 + 7*x^7;
p9(x) = 112 + 1120*x^2 + 1120*x^4 + 224*x^6 + 8*x^8;
The triangle of coefficients extended by the main diagonal with zeros starts:
[0][  0]
[1][  0,    0]
[2][  0,    1,    0]
[3][  2,    0,    2,    0]
[4][  0,    9,    0,    3,    0]
[5][  8,    0,   24,    0,    4,    0]
[6][  0,   50,    0,   50,    0,    5,   0]
[7][ 30,    0,  180,    0,   90,    0,   6,  0]
[8][  0,  245,    0,  490,    0,  147,   0,  7,  0]
[9][112,    0, 1120,    0, 1120,    0, 224,  0,  8,  0]

Cf. A109187, A290380, A298608.

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)):
A298609Row := n -> if n=0 then 0 elif n=1 then 0,0 else op(PolynomialTools:-CoefficientList(simplify(A298609Poly(n)), x)),0 fi:
seq(A298609Row(n), n=0..11);

P298609[n_] := If[n <= 1, 0, GegenbauerC[n - 1, -n, -x/2] (n - 1)/n];
Flatten[ Join[ {{0}, {0, 0}},
  Table[ Join[ CoefficientList[ P298609[n], x], {0}], {n, 2, 10}]]]

A298608(n,k) = A109187(n,k) + T(n,k).
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.
p(n, x) = Catalan(n)*(n-1)*hypergeom([1-n, -n-1], [-n+1/2], 1/2-x/4) for n >= 2.

cp's OEIS Frontend

A298300 Analog of Motzkin numbers for Coxeter type D.

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