A222006 Number of forests of rooted plane binary trees (all nodes have outdegree of 0 or 2) with n total nodes.
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
Offset: 0
Keywords
Examples
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.....
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Crossrefs
Row sums of A342770.
Programs
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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]
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
Comments