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A222006
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Number of forests of rooted plane binary trees (all nodes have outdegree of 0 or 2) with n total nodes.
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3
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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
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OFFSET
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0,4
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COMMENTS
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Here, the binary trees are sized by total number of nodes.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.....
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MAPLE
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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):
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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