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%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
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