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

%I #21 Jan 26 2023 21:42:00

%S 0,0,0,0,1,2,5,10,22,46,105,234,550,1300,3158,7740,19319,48628,123866,

%T 317954,823064,2144504,5623755,14828073,39299896,104636889,279793449,

%U 751065459,2023443031,5469566584,14830871801,40330829029,109972410220,300628862479

%N a(n) = A000055(n) - 1.

%C Number of free trees with n nodes, each node with degree <= n-2. - _Robert A. Russell_, Jan 25 2023

%H 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 a(n) = A144528(n,n-2). - _Robert A. Russell_, Jan 25 2023

%t b[n_,i_,t_,k_] := b[n,i,t,k] = If[i<1, 0, Sum[Binomial[b[i-1,i-1,k,k]

%t + j-1, j]* b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];

%t b[0,i_,t_,k_] = 1;

%t Join[{0,0,0,0,1}, Table[m = n - 3;

%t gf[x_] := 1 + Sum[b[j - 1, j - 1, m, m] x^j, {j, 1, n}];

%t ci[x_] := SymmetricGroupIndex[m + 1, x] /. x[i_] -> gf[x^i];

%t SeriesCoefficient[Series[gf[x] - (gf[x]^2 - gf[x^2])/2 + x ci[x],

%t {x, 0, n}], n], {n,5,35}]] (* _Robert A. Russell_, Jan 25 2023 *)

%Y Cf. A000055, A144528.

%K nonn

%O 0,6

%A _N. J. A. Sloane_, Dec 20 2008