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

2,2

Links

Table of n, a(n) for n=2..29.

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

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 *)

Prog

(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)))

Crossrefs

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

Sequence in context: A165993 A192312 A004080 * A027115 A077995 A276293

Adjacent sequences: A298297 A298298 A298299 * A298301 A298302 A298303

Keyword

nonn,changed

Author

F. Chapoton, Jan 16 2018

Status

approved