A121988 Number of vertices of the n-th multiplihedron.

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

Offset: 0

Author

Jonathan Vos Post, Jun 24 2007

Keywords

• easy
• nonn

Comments

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

From Peter Bala, Jan 27 2020: (Start)

This sequence is the main diagonal of the lower triangular array formed by putting the sequence [0, 1, 1, 2, 5, 14, 42, ...] of Catalan numbers (with 0 prepended) in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.

0

1 1

1 2 2

2 4 6 6

5 9 15 21 21

14 23 38 59 80 80

...

Cf. A307495.

Alternatively, the sequence can be obtained by multiplying the sequence of Catalan numbers by the array A106566. (End)

Examples

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

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, and 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 and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
• Tian-Xiao He and Louis W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra and its Applications, Volume 507, 15 October 2016, Pages 77-95.

Crossrefs

Cf. A000108, A127632, A129442, A007317, A164965, A106566, A158826, A307495.

Programs

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 *)

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 */

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..n} A106566(n,k)*A000108(k-1) with A000108(-1)=0. - Philippe Deléham, Aug 27 2007

From Vaclav Kotesovec, Oct 19 2012: (Start)

D-finite with 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).

a(n) ~ 2^(4*n-5/2)/(sqrt(Pi)*3^(n-1/2)*n^(3/2)). (End)

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

From Peter Bala, Aug 22 2024: (Start)

G.f. A(x) = 1 - 1/c(x*c(x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.

Sum_{n >= 1} a(n)*y^n = x*c(x), where y = x*(1 - x). (End)

Extensions

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