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A222006 Number of forests of rooted plane binary trees (all nodes have outdegree of 0 or 2) with n total nodes. 2
1, 1, 1, 2, 2, 4, 5, 10, 12, 27, 35, 79, 104, 244, 331, 789, 1083, 2615, 3652, 8880, 12523, 30657, 43661, 107326, 153985, 379945, 548776, 1357922, 1972153, 4892140, 7139850, 17747863, 26011843, 64776658, 95296413, 237689691, 350844814, 876313458, 1297367201, 3244521203, 4816399289 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Here, the binary trees are sized by total number of nodes.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

O.g.f.: Product_{i>=1} 1/(1 - x^i)^A126120(i-1).

a(n) ~ c * 2^n / n^(3/2), where c = 1.165663931402962361339366557... if n is even, c = 1.490999501305559555120304528... if n is odd. - Vaclav Kotesovec, Aug 31 2014

EXAMPLE

a(6) = 5: There is one forest with 6 trees, one forest with 4 trees and 3 forests with 2 trees, namely

1)...o..o..o..o..o..o...............

....................................

2)...o..o..o....o...................

.............../.\..................

..............o...o.................

....................................

3)...o........o.....................

..../.\....../.\....................

...o...o....o...o...................

....................................

4).....o....o.....5)......o.....o...

....../.\................/.\........

.....o...o..............o...o.......

..../.\..................../.\......

...o...o..................o...o.....

MAPLE

b:= proc(n) option remember; `if`(irem(n, 2)=0, 0,

      `if`(n<2, n, add(b(i)*b(n-1-i), i=1..n-2)))

    end:

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(g(n-i*j, i-2)*binomial(b(i)+j-1, j), j=0..n/i)))

    end:

a:= n-> g(n, iquo(n-1, 2)*2+1):

seq(a(n), n=0..50);  # Alois P. Heinz, Feb 26 2013

MATHEMATICA

nn=40; a=Drop[CoefficientList[Series[t=(1-(1-4x^2)^(1/2))/(2x), {x, 0, nn}], x], 1]; CoefficientList[Series[Product[1/(1-x^i)^a[[i]], {i, 1, nn-1}], {x, 0, nn}], x]

CROSSREFS

Sequence in context: A091188 A147678 A195865 * A127712 A305840 A178113

Adjacent sequences:  A222003 A222004 A222005 * A222007 A222008 A222009

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Feb 23 2013

STATUS

approved

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Last modified September 30 05:03 EDT 2020. Contains 337435 sequences. (Running on oeis4.)