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A032305 Number of rooted trees where any 2 subtrees extending from the same node have a different number of nodes. 54
1, 1, 1, 2, 3, 6, 12, 25, 51, 111, 240, 533, 1181, 2671, 6014, 13795, 31480, 72905, 168361, 393077, 914784, 2150810, 5040953, 11914240, 28089793, 66702160, 158013093, 376777192, 896262811, 2144279852, 5120176632, 12286984432, 29428496034, 70815501209 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
FORMULA
Shifts left under "EFK" (unordered, size, unlabeled) transform.
G.f.: A(x) = x*Product_{n>=1} (1+a(n)*x^n) = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Apr 07 2004
Lim_{n->infinity} a(n)^(1/n) = 2.5119824... - Vaclav Kotesovec, Nov 20 2019
G.f.: x * exp(Sum_{n>=1} Sum_{k>=1} (-1)^(k+1) * a(n)^k * x^(n*k) / k). - Ilya Gutkovskiy, Jun 30 2021
EXAMPLE
The a(6) = 6 fully unbalanced trees: (((((o))))), (((o(o)))), ((o((o)))), (o(((o)))), (o(o(o))), ((o)((o))). - Gus Wiseman, Jan 10 2018
MAPLE
A:= proc(n) if n<=1 then x else convert(series(x* (product(1+ coeff(A(n-1), x, i)*x^i, i=1..n-1)), x=0, n+1), polynom) fi end: a:= n-> coeff(A(n), x, n): seq(a(n), n=1..31); # Alois P. Heinz, Aug 22 2008
# second Maple program:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(j=0, 1, g((i-1)$2))*g(n-i*j, i-1), j=0..min(1, n/i))))
end:
a:= n-> g((n-1)$2):
seq(a(n), n=1..35); # Alois P. Heinz, Mar 04 2013
MATHEMATICA
nn=30; f[x_]:=Sum[a[n]x^n, {n, 0, nn}]; sol=SolveAlways[0 == Series[f[x]-x Product[1+a[i]x^i, {i, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}]/.sol (* Geoffrey Critzer, Nov 17 2012 *)
allnim[n_]:=If[n===1, {{}}, Join@@Function[c, Select[Union[Sort/@Tuples[allnim/@c]], UnsameQ@@(Count[#, _List, {0, Infinity}]&/@#)&]]/@IntegerPartitions[n-1]];
Table[Length[allnim[n]], {n, 15}] (* Gus Wiseman, Jan 10 2018 *)
g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0,
Sum[If[j == 0, 1, g[i-1, i-1]]*g[n-i*j, i-1], {j, 0, Min[1, n/i]}]]];
a[n_] := g[n-1, n-1];
Array[a, 35] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)
PROG
(PARI) a(n)=polcoeff(x*prod(i=1, n-1, 1+a(i)*x^i)+x*O(x^n), n)
CROSSREFS
Column k=1 of A318753.
Sequence in context: A216632 A077903 A038086 * A032218 A005829 A038087
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)