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%I #22 Aug 16 2020 08:37:16
%S 0,0,0,0,0,0,1,3,11,34,99,279,773,2103,5661,15160,40373,107355,285059,
%T 757273,2013177,5361100,14303274,38250297,102538714,275597098,
%U 742674804,2006661720,5436008057,14763754746,40196603110,109703958381,300091975184,822705857129
%N Number of noncaterpillar trees on n nodes (A000055-A005418).
%H Alois P. Heinz, <a href="/A052471/b052471.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CaterpillarGraph.html">Caterpillar Graph</a>
%p with(numtheory):
%p b:= proc(n) option remember; `if`(n<=1, n,
%p (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
%p end:
%p a:= n-> b(n) -(add(b(k) *b(n-k), k=0..n)-`if`(irem(n, 2)=0,
%p b(n/2), 0))/2 -ceil(2^(n-4) + 2^(iquo(n-2, 2)-1)):
%p seq(a(n), n=1..40); # _Alois P. Heinz_, May 18 2013
%t b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n-1}])/(n-1)]; a[n_] := b[n] - (Sum[b[k]*b[n-k], {k, 0, n}] - If[ Mod[n, 2] == 0, b[n/2], 0])/2 - Ceiling[2^(n-4) + 2^(Quotient[n-2, 2] - 1)]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Feb 19 2016, after _Alois P. Heinz_ *)
%Y Cf. A000055, A005418.
%K nonn
%O 1,8
%A _Eric W. Weisstein_
%E a(14) and up from _Eric W. Weisstein_, Jul 17 2004.