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A121988 Number of vertices of the n-th multiplihedron. 7
0, 1, 2, 6, 21, 80, 322, 1348, 5814, 25674, 115566, 528528, 2449746, 11485068, 54377288, 259663576, 1249249981, 6049846848, 29469261934, 144293491564, 709806846980, 3506278661820, 17385618278700, 86500622296800, 431718990188850, 2160826237261692 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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, Jun 13 2008

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..500

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5, pp. 1-24.

David Callan, A combinatorial interpretation of the Catalan transform of the Catalan numbers, arXiv:1111.0996 [math.CO], 2011.

Stefan Forcey, Convex Hull Realizations of the Multiplihedra, Theorem 3.2, p. 8, arXiv:0706.3226 [math.AT], 2007-2008.

FORMULA

a(0) = 0; a(n) = C(n-1) + Sum_{i=1..(n-1)} a(i)*a(n-i), where C(n) = A000108(n).

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 Deléham, Aug 27 2007

Recurrence: 3*(n-1)*n*a(n) = 14*(n-1)*(2*n-3)*a(n-1) - 4*(4*n-9)*(4*n-7)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 2^(4*n-5/2)/(sqrt(Pi)*3^(n-1/2)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

G.f.: A(x) satisfies A(x)=x*(1+A(x))/((1-A(x))*(1+A(x)^3). - Vladimir Kruchinin, Jun 01 2014.

G.f. is series reversion of (x - x^2) * (1 - x + x^2) = x - 2*x^2 + 2*x^3 - x^4. - Michael Somos, Jun 01 2014

EXAMPLE

G.f. = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 + 1348*x^7 + 5814*x^8 + ...

MAPLE

a:= proc(n) option remember; `if`(n<3, n, (14*(n-1)*(2*n-3)*a(n-1)

      -4*(4*n-9)*(4*n-7)*a(n-2))/ (3*n*(n-1)))

    end:

seq(a(n), n=0..30);  # Alois P. Heinz, Oct 20 2012

MATHEMATICA

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}] (* Robert G. Wilson v, Jun 28 2007 *)

a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ x - 2 x^2 + 2 x^3 - x^4, {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Jun 01 2014 *)

a[0] = 0; a[n_] := Binomial[2n-2, n-1]*Hypergeometric2F1[1/2, 1-n, 2-2n, 4] /n; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 31 2016 *)

PROG

(PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( x - 2*x^2 + 2*x^3 - x^4 + x * O(x^n)), n))}; /* Michael Somos, Jun 01 2014 */

CROSSREFS

Cf. A000108, A129442, A007317, A164965.

Sequence in context: A106228 A150204 A129442 * A150205 A150206 A150207

Adjacent sequences:  A121985 A121986 A121987 * A121989 A121990 A121991

KEYWORD

easy,nonn,changed

AUTHOR

Jonathan Vos Post, Jun 24 2007

EXTENSIONS

More terms from Robert G. Wilson v, Jun 28 2007

Replaced arXiv URL by non-cached version - R. J. Mathar, Oct 23 2009

STATUS

approved

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Last modified December 7 14:52 EST 2016. Contains 278877 sequences.