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%I M2803 N1128 #46 Mar 23 2023 23:09:28
%S 1,3,9,26,75,214,612,1747,4995,14294,40967,117560,337830,972027,
%T 2800210,8075889,23315775,67380458,194901273,564239262,1634763697,
%U 4739866803,13752309730,39926751310,115988095896,337138003197
%N Number of trees with n nodes, 2 of which are labeled.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000243/b000243.txt">Table of n, a(n) for n=2..200</a>
%H R. J. Mathar, <a href="http://arxiv.org/abs/1603.00077">Topologically distinct sets of non-intersecting circles in the plane</a>, arXiv:1603.00077 [math.CO] (2016), Table 5.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F a(n) = A000107(n) - A000081(n). - _Christian G. Bower_, Nov 15 1999
%F G.f.: A(x) = B(x)^2/(1-B(x)), where B(x) is g.f. for rooted trees with n nodes, cf. A000081. - _Vladeta Jovovic_, Oct 19 2001
%F a(n) = A000106(n) + A304068(n). - _Brendan McKay_, May 05 2018
%p b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-1)^2/(1-B(n-1)), x=0, n+1), x,n): seq(a(n), n=2..27); # _Alois P. Heinz_, Aug 21 2008
%t b[n_] := b[n] = If[ n <= 1 , n, Sum[k*b[k]*s[n - 1, k], {k, 1, n - 1}]/(n - 1) ]; s[n_, k_] := s[n, k] = Sum[ b[n + 1 - j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[ b[k]*x^k, {k, 1, n}]; a[n_] := Coefficient[ Series[ B[n - 1]^2/(1 - B[n - 1]), {x, 0, n + 1}], x, n]; Table[ a[n], {n, 2, 27}] (* _Jean-François Alcover_, Jan 25 2012, translated from Maple *)
%Y Column k=2 of A034799.
%K nonn,easy,nice
%O 2,2
%A _N. J. A. Sloane_
%E More terms and new description from _Christian G. Bower_, Nov 15 1999
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