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A330951 Number of singleton-reduced unlabeled rooted trees with n nodes.

%I A330951 #12 Nov 16 2021 07:10:17
%S A330951 1,1,1,3,5,11,24,52,119,272,635,1499,3577,8614,20903,51076,125565,
%T A330951 310302,770536,1921440,4809851,12081986,30445041,76938794,194950040,
%U A330951 495174037,1260576786,3215772264,8219437433,21046602265,53982543827,138678541693,356785641107
%N A330951 Number of singleton-reduced unlabeled rooted trees with n nodes.
%C A330951 A rooted tree is singleton-reduced if no non-leaf node has all singleton branches, where a rooted tree is a singleton if its root has degree 1.
%H A330951 Andrew Howroyd, <a href="/A330951/b330951.txt">Table of n, a(n) for n = 1..1000</a>
%F A330951 G.f.: A(x) satisfies A(x) = x + x*exp(Sum_{k>=1} A(x^k)/k) - x*exp(Sum_{k>=1} x^k*A(x^k)/(1 + x^k)/k). - _Andrew Howroyd_, Dec 10 2020
%F A330951 a(n) ~ c * d^n / n^(3/2), where d = 2.69474016697407303512228736537683134987637576... and c = 0.41800971384719166056172258174139385922545... - _Vaclav Kotesovec_, Nov 16 2021
%e A330951 The a(1) = 1 through a(6) = 11 trees:
%e A330951   o  (o)  (oo)  (ooo)   (oooo)    (ooooo)
%e A330951                 ((oo))  ((ooo))   ((oooo))
%e A330951                 (o(o))  (o(oo))   (o(ooo))
%e A330951                         (oo(o))   (oo(oo))
%e A330951                         ((o(o)))  (ooo(o))
%e A330951                                   ((o)(oo))
%e A330951                                   ((o(oo)))
%e A330951                                   ((oo(o)))
%e A330951                                   (o((oo)))
%e A330951                                   (o(o)(o))
%e A330951                                   (o(o(o)))
%t A330951 urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
%t A330951 Table[Length[Select[urt[n],FreeQ[#,q:{__List}/;Times@@Length/@q==1]&]],{n,10}]
%o A330951 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A330951 seq(n)={my(v=vector(n)); v[1]=1; for(n=1, #v-1, v[n+1] = EulerT(v[1..n])[n] - EulerT(Vec(x^2*Ser(v[1..n-1])/(1+x), -n))[n]); v} \\ _Andrew Howroyd_, Dec 10 2020
%Y A330951 The Matula-Goebel numbers of these trees are given by A330943.
%Y A330951 The series-reduced case is A001678.
%Y A330951 Unlabeled rooted trees are counted by A000081.
%Y A330951 Singleton-reduced phylogenetic trees are A000311.
%Y A330951 Cf. A000669, A003238, A004111, A324694.
%K A330951 nonn
%O A330951 1,4
%A A330951 _Gus Wiseman_, Jan 15 2020
%E A330951 Terms a(19) and beyond from _Andrew Howroyd_, Dec 10 2020