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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 Alois P. Heinz, Table of n, a(n) for n = 1..1000 Gus Wiseman, Illustration of the first 8 terms of A032305. 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 Cf. A000081, A001678, A003238, A004111, A213920, A273873, A290689, A291443, A297571. Column k=1 of A318753. Sequence in context: A216632 A077903 A038086 * A032218 A005829 A038087 Adjacent sequences:  A032302 A032303 A032304 * A032306 A032307 A032308 KEYWORD nonn AUTHOR STATUS approved

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Last modified May 17 14:19 EDT 2022. Contains 353746 sequences. (Running on oeis4.)