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

Author

F. Chapoton, Jan 16 2018

Keywords

• nonn

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Sage

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)))

Formula

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