0,3

The number of facets of the n-th multiplihedron is (n*(n-1)/2) + (2^(n-1)) -1, as proved in Forcey, Theorem 2.1, p. 4. Abstract: "We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the n^{th} polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes."

G.f. = x*c(x)*c(x*c(x)) where c(x) is the generating function of the Catalan numbers C(n). Thus a(n) is the Catalan transform of the sequence C(n-1). Reference for the definition of Catalan transform is the paper by Paul Barry. - Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007

A129442 is an essentially identical sequence. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 13 2008

Paul Barry, A Catalan transform and related transformations on integer sequences, Journal of Integer Sequences, Vol. 8 (2005), pp. 1-24.

Stefan Forcey, Convex Hull Realizations of the Multiplihedra, Theorem 3.2, p. 8.

a(0) = 0; a(n) = C(n-1) + SUM[i=1..(n-1)]a(i)*a(n-i), where C(n) is the Catalan sequence A000108.

G.f. = (1-sqrt(2*sqrt(1-4x)-1))/2. a(n) = (1/n)*Sum_{k=1..n}(binomial(2*n-k-1,n-1)*binomial(2k-2, k-1)); a(0)=0. - Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007

a(n)=Sum_{k, 0<=k<=n}A106566(n,k)*A000108(k-1) with A000108(-1)=0 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 27 2007

a[0] = 0; a[n_] := a[n] = (2 n - 2)!/((n - 1)! n!) + Sum[ a[i]*a[n - i], {i, n - 1}]; Table[ a@n, {n, 0, 24}] (* RGWV *)

Cf. A000108.

Cf. A129442, A007317.

Sequence in context: A112806 A106223 A106228 this_sequence A129442 A032347 A032346

Adjacent sequences: A121985 A121986 A121987 this_sequence A121989 A121990 A121991

easy,nonn

Jonathan Vos Post (jvospost2(AT)yahoo.com), Jun 24 2007

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 28 2007