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 A198518 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x^n)/(1+x^n) * x^n/n  ). 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 David Callan, Rooted trees with no out-degree = 1, (7-July-2014). David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014). FORMULA 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 a(n) = A001678(n + 1) + A001678(n + 2). - Gus Wiseman, Jan 22 2020 Euler transform of A001678(n + 1). - Gus Wiseman, Feb 03 2020 EXAMPLE 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, ...] (this is, up to offset, A001678), 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*...). From Joerg Arndt, Jun 28 2014: (Start) 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) From Gus Wiseman, Jan 22 2020: (Start) 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) MAPLE 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: seq(a(n), n=0..50);  # Alois P. Heinz, Jul 02 2014 MATHEMATICA 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]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 21 2017, after Alois P. Heinz *) 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 *) PROG (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)} CROSSREFS Cf. A052855, A246403. The labeled version is A254382. 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. Cf. A000669, A004111, A108919, A291636, A330951, A331488, A331934. Sequence in context: A000049 A000050 A050253 * A182558 A298204 A265581 Adjacent sequences:  A198515 A198516 A198517 * A198519 A198520 A198521 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 26 2011 STATUS approved

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Last modified April 16 04:49 EDT 2021. Contains 343030 sequences. (Running on oeis4.)