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A121988
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Number of vertices of the n-th multiplihedron.
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10
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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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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
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
...
Alternatively, the sequence can be obtained by multiplying the above sequence of Catalan numbers by the array A106566.
(End)
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LINKS
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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).
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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) = 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
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). - 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
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EXAMPLE
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G.f. = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 + 1348*x^7 + 5814*x^8 + ...
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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:
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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, 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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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