cp's OEIS Frontend

A222006 Number of forests of rooted plane binary trees (all nodes have outdegree of 0 or 2) with n total nodes.

%I A222006 #37 Sep 23 2021 06:22:30
%S A222006 1,1,1,2,2,4,5,10,12,27,35,79,104,244,331,789,1083,2615,3652,8880,
%T A222006 12523,30657,43661,107326,153985,379945,548776,1357922,1972153,
%U A222006 4892140,7139850,17747863,26011843,64776658,95296413,237689691,350844814,876313458,1297367201,3244521203,4816399289
%N A222006 Number of forests of rooted plane binary trees (all nodes have outdegree of 0 or 2) with n total nodes.
%C A222006 Here, the binary trees are sized by total number of nodes.
%H A222006 Alois P. Heinz, <a href="/A222006/b222006.txt">Table of n, a(n) for n = 0..1000</a>
%F A222006 O.g.f.: Product_{i>=1} 1/(1 - x^i)^A126120(i-1).
%F A222006 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 A222006 a(6) = 5: There is one forest with 6 trees, one forest with 4 trees and 3 forests with 2 trees, namely
%e A222006 1)...o..o..o..o..o..o...............
%e A222006 ....................................
%e A222006 2)...o..o..o....o...................
%e A222006 .............../.\..................
%e A222006 ..............o...o.................
%e A222006 ....................................
%e A222006 3)...o........o.....................
%e A222006 ..../.\....../.\....................
%e A222006 ...o...o....o...o...................
%e A222006 ....................................
%e A222006 4).....o....o.....5)......o.....o...
%e A222006 ....../.\................/.\........
%e A222006 .....o...o..............o...o.......
%e A222006 ..../.\..................../.\......
%e A222006 ...o...o..................o...o.....
%p A222006 b:= proc(n) option remember; `if`(irem(n, 2)=0, 0,
%p A222006       `if`(n<2, n, add(b(i)*b(n-1-i), i=1..n-2)))
%p A222006     end:
%p A222006 g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A222006       add(g(n-i*j, i-2)*binomial(b(i)+j-1, j), j=0..n/i)))
%p A222006     end:
%p A222006 a:= n-> g(n, iquo(n-1, 2)*2+1):
%p A222006 seq(a(n), n=0..50);  # _Alois P. Heinz_, Feb 26 2013
%t A222006 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 A222006 Row sums of A342770.
%K A222006 nonn
%O A222006 0,4
%A A222006 _Geoffrey Critzer_, Feb 23 2013