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%I M3014 N1220 #32 Mar 23 2023 23:09:23
%S 3,16,67,251,888,3023,10038,32722,105228,334836,1056611,3311784,
%T 10322791,32026810,98974177,304835956,936147219,2867586542,8764280567,
%U 26733395986,81399821915,247459136331,751211286356,2277496842016
%N Number of trees with n nodes, 3 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="/A000269/b000269.txt">Table of n, a(n) for n=3..200</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F G.f.: A(x) = B(x)^3*(3-2*B(x))/(1-B(x))^3, where B(x) is g.f. for rooted trees with n nodes, cf. A000081. - _Vladeta Jovovic_, Oct 19 2001
%F a(n) = A000524(n) - 2*A000243(n).
%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_] := SeriesCoefficient[ B[n-1]^3 * (2*B[n-1]-3) / (B[n-1]-1)^3, {x, 0, n}]; Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, Jan 27 2015 *)
%Y Column k=3 of A034799.
%K nonn,easy,nice
%O 3,1
%A _N. J. A. Sloane_
%E More terms, new description and formula from _Christian G. Bower_, Nov 15 1999
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