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A198518
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G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x^n)/(1+x^n) * x^n/n ).
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13
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1, 1, 1, 2, 3, 5, 9, 16, 29, 54, 102, 194, 375, 730, 1434, 2837, 5650, 11311, 22767, 46023, 93422, 190322, 389037, 797613, 1639878, 3380099, 6983484, 14459570, 29999618, 62357426, 129843590, 270807835, 565674584, 1183301266, 2478624060, 5198504694, 10916110768, 22948299899
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OFFSET
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0,4
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COMMENTS
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For n>=1, a(n) is the number of rooted trees (see A000081) with n non-root nodes where non-root nodes cannot have out-degree 1, see the note by David Callan and the example. Imposing the condition also for the root node gives A001678. - Joerg Arndt, Jun 28 2014
Compare definition to G(x) = exp( Sum_{n>=1} G(x^n)*x^n/n ), where G(x) is the g.f. of A000081, the number of rooted trees with n nodes.
Number of forests of lone-child-avoiding rooted trees with n unlabeled vertices. - Gus Wiseman, Feb 03 2020
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LINKS
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FORMULA
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Euler transform of coefficients in A(x)/(1+x), where g.f. A(x) = Sum_{n>=0} a(n)*x^n.
a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.18946198566085056388702757711..., c = 1.3437262442171062526771597... . - Vaclav Kotesovec, Sep 03 2014
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 16*x^7 + 29*x^8 +...
where
log(A(x)) = A(x)/(1+x)*x + A(x^2)/(1+x^2)*x^2/2 + A(x^3)/(1+x^3)*x^3/3 +...
The coefficients in A(x)/(1+x) begin:
[1, 0, 1, 1, 2, 3, 6, 10, 19, 35, 67, 127, 248, 482, 952, 1885, 3765, ...]
from which g.f. A(x) may be generated by the Euler transform:
A(x) = 1/((1-x)^1*(1-x^2)^0*(1-x^3)^1*(1-x^4)^1*(1-x^5)^2*(1-x^6)^3*(1-x^7)^6*(1-x^8)^10*(1-x^9)^19*(1-x^10)^35*...).
The a(6) = 9 rooted trees with 6 non-root nodes as described in the comment are:
: level sequence out-degrees (dots for zeros)
: 1: [ 0 1 2 3 3 3 2 ] [ 1 2 3 . . . . ]
: O--o--o--o
: .--o
: .--o
: .--o
:
: 2: [ 0 1 2 3 3 2 2 ] [ 1 3 2 . . . . ]
: O--o--o--o
: .--o
: .--o
: .--o
:
: 3: [ 0 1 2 3 3 2 1 ] [ 2 2 2 . . . . ]
: O--o--o--o
: .--o
: .--o
: .--o
:
: 4: [ 0 1 2 2 2 2 2 ] [ 1 5 . . . . . ]
: O--o--o
: .--o
: .--o
: .--o
: .--o
:
: 5: [ 0 1 2 2 2 2 1 ] [ 2 4 . . . . . ]
: O--o--o
: .--o
: .--o
: .--o
: .--o
:
: 6: [ 0 1 2 2 2 1 1 ] [ 3 3 . . . . . ]
: O--o--o
: .--o
: .--o
: .--o
: .--o
:
: 7: [ 0 1 2 2 1 2 2 ] [ 2 2 . . 2 . . ]
: O--o--o
: .--o
: .--o--o
: .--o
:
: 8: [ 0 1 2 2 1 1 1 ] [ 4 2 . . . . . ]
: O--o--o
: .--o
: .--o
: .--o
: .--o
:
: 9: [ 0 1 1 1 1 1 1 ] [ 6 . . . . . . ]
: O--o
: .--o
: .--o
: .--o
: .--o
: .--o
(End)
The a(0) = 1 through a(6) = 9 rooted trees with n + 1 nodes where non-root vertices cannot have out-degree 1:
o (o) (oo) (ooo) (oooo) (ooooo) (oooooo)
((oo)) ((ooo)) ((oooo)) ((ooooo))
(o(oo)) (o(ooo)) (o(oooo))
(oo(oo)) (oo(ooo))
((o(oo))) (ooo(oo))
((o(ooo)))
((oo)(oo))
((oo(oo)))
(o(o(oo)))
(End)
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MAPLE
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with(numtheory):
b:= proc(n) b(n):= `if`(n=0, 1, a(n)-b(n-1)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*b(d-1), d=divisors(j))*a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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b[n_] := b[n] = If[n==0, 1, a[n] - b[n-1]];
a[n_] := a[n] = If[n==0, 1, Sum[Sum[d*b[d-1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]], {ptn, IntegerPartitions[n-1]}];
Table[Length[Select[urt[n], FreeQ[Z@@#, {_}]&]], {n, 10}] (* Gus Wiseman, Jan 22 2020 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, subst(A/(1+x), x, x^m+x*O(x^n))*x^m/m))); polcoeff(A, n)}
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CROSSREFS
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Unlabeled rooted trees are A000081.
Lone-child-avoiding rooted trees are A001678(n+1).
Topologically series-reduced rooted trees are A001679.
Labeled lone-child-avoiding rooted trees are A060356.
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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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