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A121988
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Number of vertices of the n-th multiplihedron.
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6
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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Paul Barry, A Catalan transform and related transformations on integer sequences, Journal of Integer Sequences, Vol. 8 (2005), pp. 1-24.
D. Callan, A combinatorial interpretation of the Catalan transform of the Catalan numbers, arXiv:1111.0996
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..500
Stefan Forcey, Convex Hull Realizations of the Multiplihedra, Theorem 3.2, p. 8.
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FORMULA
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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, 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
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A000108.
Cf. A129442, A007317.
Sequence in context: A106228 A150204 A129442 * A150205 A150206 A150207
Adjacent sequences: A121985 A121986 A121987 * A121989 A121990 A121991
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post, Jun 24 2007
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EXTENSIONS
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More terms from Robert G. Wilson v, Jun 28 2007
Replaced arXiv URL by non-cached version - R. J. Mathar, Oct 23 2009
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STATUS
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approved
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