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Number of forests of rooted plane binary trees (all nodes have outdegree of 0 or 2) with n total nodes.
3

%I #37 Sep 23 2021 06:22:30

%S 1,1,1,2,2,4,5,10,12,27,35,79,104,244,331,789,1083,2615,3652,8880,

%T 12523,30657,43661,107326,153985,379945,548776,1357922,1972153,

%U 4892140,7139850,17747863,26011843,64776658,95296413,237689691,350844814,876313458,1297367201,3244521203,4816399289

%N Number of forests of rooted plane binary trees (all nodes have outdegree of 0 or 2) with n total nodes.

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

%H Alois P. Heinz, <a href="/A222006/b222006.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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

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

%e 1)...o..o..o..o..o..o...............

%e ....................................

%e 2)...o..o..o....o...................

%e .............../.\..................

%e ..............o...o.................

%e ....................................

%e 3)...o........o.....................

%e ..../.\....../.\....................

%e ...o...o....o...o...................

%e ....................................

%e 4).....o....o.....5)......o.....o...

%e ....../.\................/.\........

%e .....o...o..............o...o.......

%e ..../.\..................../.\......

%e ...o...o..................o...o.....

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

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

%p end:

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

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

%p end:

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

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Feb 26 2013

%t 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]

%Y Row sums of A342770.

%K nonn

%O 0,4

%A _Geoffrey Critzer_, Feb 23 2013