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A144520 a(n) = A000055(n) - 1.

%I A144520 #21 Jan 26 2023 21:42:00
%S A144520 0,0,0,0,1,2,5,10,22,46,105,234,550,1300,3158,7740,19319,48628,123866,
%T A144520 317954,823064,2144504,5623755,14828073,39299896,104636889,279793449,
%U A144520 751065459,2023443031,5469566584,14830871801,40330829029,109972410220,300628862479
%N A144520 a(n) = A000055(n) - 1.
%C A144520 Number of free trees with n nodes, each node with degree <= n-2. - _Robert A. Russell_, Jan 25 2023
%H A144520 Rebecca Neville, <a href="https://web.archive.org/web/20191029092609/http://gtn.kazlow.info:80/GTN54.pdf">Graphs whose vertices are forests with bounded degree</a>, Graph Theory Notes of New York, LIV (2008), 12-21. [Wayback Machine link]
%F A144520 a(n) = A144528(n,n-2). - _Robert A. Russell_, Jan 25 2023
%t A144520 b[n_,i_,t_,k_] := b[n,i,t,k] = If[i<1, 0, Sum[Binomial[b[i-1,i-1,k,k]
%t A144520   + j-1, j]* b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
%t A144520   b[0,i_,t_,k_] = 1;
%t A144520 Join[{0,0,0,0,1}, Table[m = n - 3;
%t A144520   gf[x_] := 1 + Sum[b[j - 1, j - 1, m, m] x^j, {j, 1, n}];
%t A144520   ci[x_] := SymmetricGroupIndex[m + 1, x] /. x[i_] -> gf[x^i];
%t A144520   SeriesCoefficient[Series[gf[x] - (gf[x]^2 - gf[x^2])/2 + x ci[x],
%t A144520 {x, 0, n}], n], {n,5,35}]] (* _Robert A. Russell_, Jan 25 2023 *)
%Y A144520 Cf. A000055, A144528.
%K A144520 nonn
%O A144520 0,6
%A A144520 _N. J. A. Sloane_, Dec 20 2008