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

3,2

Comments

See proposition 3.3 of the Athanasiadis-Savvidou reference.

Links

Table of n, a(n) for n=3..30.

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.

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

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 *)

Prog

(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))

Crossrefs

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

Sequence in context: A291264 A171849 A199937 * A003212 A156945 A006817

Adjacent sequences: A290377 A290378 A290379 * A290381 A290382 A290383

Keyword

nonn

Author

F. Chapoton, Jul 28 2017

Status

approved