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%I M4843 N2070 #41 Mar 24 2017 00:47:48
%S 12,60,210,630,1736,4536,11430,28050,67452,159588,372554,859950,
%T 1965840,4456176,10026702,22412970,49806980,110100060,242220594,
%U 530578950,1157627352,2516581800,5452594550,11777604930,25367149836,54492396756,116769422490,249644973150
%N Number of labeled trees of diameter 3 with n nodes.
%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 Colin Barker, <a href="/A000554/b000554.txt">Table of n, a(n) for n = 4..1000</a>
%H J. Riordan, <a href="http://dx.doi.org/10.1147/rd.45.0473">Enumeration of trees by height and diameter</a>, IBM J. Res. Dev. 4 (1960), 473-478.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (9,-33,63,-66,36,-8).
%F a(n) = n(n-1)*S2(n-2, 2) where S2(n, k) denotes the Stirling numbers of 2nd kind. - Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001
%F a(n) = n*(n-1)*(2^(n-3) - 1) = 2*A000217(n-1)*A000225(n-3). - _Robert G. Wilson v_, Jul 01 2007, corrected by _Ilya Gutkovskiy_, Sep 17 2016
%F a(n) = Sum_{k=1..n-3} binomial(n,2)*binomial(n-2,k). The sum gives the number of Prüfer sequences with exactly 2 distinct digits. - _Geoffrey Critzer_, Sep 17 2016
%F E.g.f.: (x*(exp(x)-1))^2/2. - _Geoffrey Critzer_, Sep 17 2016
%F O.g.f.: 2*x^4*(6 - 24*x + 33*x^2 - 18*x^3 + 4*x^4)/((1 - x)^3*(1 - 2*x)^3). - _Ilya Gutkovskiy_, Sep 17 2016
%F a(n) = (2^n-8)*(n-1)*n/8. - _Colin Barker_, Sep 18 2016
%t f[n_] := n (n - 1)*StirlingS2[n - 2, 2]; Table[ f@n, {n, 4, 29}] (* _Robert G. Wilson v_, Jul 01 2007 *)
%o (PARI) Vec(2*x^4*(6-24*x+33*x^2-18*x^3+4*x^4)/((1-x)^3*(1-2*x)^3) + O(x^40)) \\ _Colin Barker_, Sep 18 2016
%K nonn,easy
%O 4,1
%A _N. J. A. Sloane_
%E More terms from _Robert G. Wilson v_, Jul 01 2007
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