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A121686 Number of branches in all binary trees with n edges. A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child. 3
2, 6, 22, 84, 324, 1254, 4862, 18876, 73372, 285532, 1112412, 4338536, 16938120, 66192390, 258909390, 1013586540, 3971224620, 15571021620, 61096813140, 239888764440, 942483155640, 3705043827420, 14573172387852, 57351122857944 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..500

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

FORMULA

a(n) = Sum_{k=1..n} k*A121685(n,k).

G.f.: (1 - 2*z) * (1 - 3*z - (1 - z)*sqrt(1 - 4*z))/(z^2*sqrt(1 - 4*z)).

Recurrence: (n + 2)*(n^2 - 2*n + 3)*a(n) = 2*(2*n - 1)*(n^2 + 2)*a(n-1). - Vaclav Kotesovec, Dec 10 2013

a(n) = 2*(n^2 + 2)*binomial(2*n, n)/((n + 1)*(n + 2)). - Vaclav Kotesovec, Dec 10 2013

EXAMPLE

a(1) = 2 because we have two binary trees with 1 edge, namely / and \, with a total of 2 branches.

MAPLE

G:=(1-2*z)*(1-3*z-(1-z)*sqrt(1-4*z))/z^2/sqrt(1-4*z): Gser:=series(G, z=0, 31): seq(coeff(Gser, z, n), n=1..27);

CROSSREFS

Cf. A121685.

Sequence in context: A150243 A200316 A164870 * A245904 A128723 A150244

Adjacent sequences:  A121683 A121684 A121685 * A121687 A121688 A121689

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Aug 15 2006

STATUS

approved

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Last modified June 20 02:46 EDT 2021. Contains 345154 sequences. (Running on oeis4.)