A290380 Analog of Motzkin sums for Coxeter type D.

1, 4, 12, 36, 105, 306, 889, 2584, 7515, 21880, 63778, 186132, 543855, 1590876, 4658580, 13655472, 40065243, 117654876, 345786396, 1017040380, 2993498739, 8816790906, 25984489545, 76625467128, 226085062525, 667415280376, 1971209865654, 5824651789852

Offset: 3

Author

F. Chapoton, Jul 28 2017

Keywords

• nonn

Comments

See proposition 3.3 of the Athanasiadis-Savvidou reference.

Links

• Christos A. Athanasiadis and Christina Savvidou, The Local h-Vector of the Cluster Subdivision of a Simplex, Séminaire Lotharingien de Combinatoire 66 (2012), Article B66c.

Crossrefs

Cf. A001006, A005043 (type A), A246437 (type B).

Programs

Maple

a:= proc(n) option remember; `if`(n<5, [0$2, 1, 4][n],
      ((n-2)*(2*n-3)*(n-4)*a(n-1)+3*(n-2)*(n-3)^2*
      a(n-2))/((n-3)*(n-4)*n))
    end:
seq(a(n), n=3..35);  # Alois P. Heinz, Jul 28 2017

Mathematica

Table[Sum[(n - 2)/i*Binomial[2i - 2, i - 1] Binomial[n - 2, 2i - 2], {i, n/2}], {n, 3, 50}] (* Indranil Ghosh, Jul 29 2017 *)
a[n_] := (n - 2) Hypergeometric2F1[1 - n/2, 3/2 - n/2, 2, 4];
Table[a[n], {n, 3, 30}] (* Peter Luschny, Jan 23 2018 *)

Sage

def A290380(n):
    return sum(ZZ(n - 2) / i * binomial(2 * i - 2, i - 1) *
               binomial(n - 2, 2 * i - 2)
               for i in range(1, n // 2 + 1))

Formula

a(n) = Sum_{i=1..n/2} (n-2)/i*binomial(2*i-2, i-1)*binomial(n-2, 2*i-2).

From Peter Luschny, Jan 23 2018: (Start)

a(n) = (n - 2)*hypergeom([1 - n/2, 3/2 - n/2], [2], 4).

a(n) = (-1)^n (n - 2)*hypergeom([2 - n, 3/2], [3], 4).

a(n) = (n - 2)*A001006(n-2). (End)

G.f.: ((x-2)*sqrt(-3*x^2-2*x+1)-3*x^2-3*x+2)/(2*sqrt(-3*x^2-2*x+1)). - Vladimir Kruchinin, Jun 21 2024