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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 R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004. See p. 19. 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. Stefan Forcey, Aaron Lauve, Frank Sottile, New Hopf Structures on Binary Trees, dmtcs:2740 - Discrete Mathematics & Theoretical Computer Science, January 1, 2009, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019. Tian-Xiao He, Louis W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra and its Applications, Volume 507, 15 October 2016, Pages 77-95. 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; 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; 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 STATUS approved

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Last modified October 15 01:40 EDT 2019. Contains 328025 sequences. (Running on oeis4.)