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1, 2, 9, 44, 225, 1182, 6321, 34232, 187137, 1030490, 5707449, 31760676, 177435297, 994551222, 5590402785, 31500824304, 177880832001, 1006362234162, 5703029112297, 32367243171740, 183945502869345, 1046646207221582, 5961966567317649, 33995080211156904
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OFFSET
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0,2
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COMMENTS
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Central coefficients T(2n,n) of the Riordan array ((1-x)/(1-2x), x(1-x)/(1-2x)), A105306.
a(n) counts the bi-degree sequences of directed trees (i.e., digraphs whose underlying graph is a tree) with n edges. - Nikos Apostolakis, Dec 31 2016
a(n) is also the number of Dyck paths having exactly n peaks in level 1 and n peaks in level 2 and no other peaks. a(2) = 9: /\/\//\/\\, /\//\/\\/\, //\/\\/\/\, /\/\//\\//\\, /\//\\/\//\\, /\//\\//\\/\, //\\/\/\//\\, //\\/\//\\/\, //\\//\\/\/\. - Alois P. Heinz, Jun 20 2017
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
V. V. Kruchinin and D. V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, arXiv preprint arXiv:1206.0877 [math.CO], 2012, and J. Int. Seq. 15 (2012) #12.9.3
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FORMULA
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E.g.f.: 1+exp(3*x)*Bessel_I(1,2*sqrt(2)*x)/sqrt(2) +int(exp(3*x) *Bessel_I(1,2*sqrt(2)*x) /(sqrt(2)*x),x).
G.f.: 1/4 - (x-3)/(4*sqrt(x^2-6*x+1)). - Dmitry Kruchinin, Aug 31 2012
Conjecture: n*(n-1)*a(n) -3*(2*n-1)*(n-1)*a(n-1) +n*(n-2)*a(n-2) = 0. - R. J. Mathar, Dec 03 2014
a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(n+k,n). - Nikos Apostolakis, Dec 31 2016
a(n) = (n+1)*hypergeom([1-n, -n], [2], 2). - Peter Luschny, Jan 02 2017
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MAPLE
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a:= proc(n) option remember; `if`(n<2, n+1,
(6*n-3)/n*a(n-1) -(n-2)/(n-1)*a(n-2))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Jun 22 2017
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MATHEMATICA
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a[n_] := Sum[Binomial[n - 1, k - 1]*Binomial[n + k, n], {k, 0, n}]; Array[a, 25, 0] (* or *)
CoefficientList[ Series[1/4 - (x - 3)/(4 Sqrt[x^2 - 6x +1]), {x, 0, 25}], x] (* Robert G. Wilson v, Dec 31 2016 *)
Table[(n+1)Hypergeometric2F1[1-n, -n, 2, 2], {n, 0, 21}] (* Peter Luschny, Jan 02 2017 *)
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CROSSREFS
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Cf. A001003, A105306.
Row n=2 of A288972.
Sequence in context: A013981 A216861 A199308 * A162356 A339440 A026302
Adjacent sequences: A176476 A176477 A176478 * A176480 A176481 A176482
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Apr 18 2010
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STATUS
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approved
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