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%I M5086 N2202 #37 Apr 13 2022 13:25:15
%S 20,74,186,388,721,1236,1995,3072,4554,6542,9152,12516,16783,22120,
%T 28713,36768,46512,58194,72086,88484,107709,130108,156055,185952,
%U 220230,259350,303804,354116,410843,474576,545941,625600,714252,812634,921522,1041732,1174121,1319588,1479075,1653568,1844098,2051742,2277624,2522916,2788839,3076664,3387713,3723360,4085032,4474210
%N Powers of rooted tree enumerator.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
%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 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%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>
%p A000529:=(z-2)*(3*z**3-12*z**2+18*z-10)/(z-1)**6; [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]
%p a:= n-> (Matrix([[0,-3,0,3,4,4]]). Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [6,-15,20, -15,6,-1][i] else 0 fi)^n)[1,1]: seq(a(n), n=1..24); # _Alois P. Heinz_, Aug 26 2008
%t a[n_] := ({0, -3, 0, 3, 4, 4}.MatrixPower[Table[If[i == j-1, 1, If[j == 1, {6, -15, 20, -15, 6, -1}[[i]], 0]], {i, 1, 6}, {j, 1, 6}], n])[[1]]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Oct 14 2014, after _Alois P. Heinz_ *)
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, Nov 14 2010
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