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%I M1415 N0553
%S 1,2,5,12,30,74,188,478,1235,3214,8450,22370,59676,160140,432237,
%T 1172436,3194870,8741442,24007045,66154654,182864692,506909562,
%U 1408854940,3925075510,10959698606,30665337738,85967279447,241433975446,679192039401,1913681367936,5399924120339
%N 2nd power of rooted tree enumerator; number of linear forests of 2 rooted trees.
%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 T. D. Noe and Alois P. Heinz, <a href="/A000106/b000106.txt">Table of n, a(n) for n = 2..1000</a> (terms n = 2..200 from T. D. Noe)
%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&service=Search&searchTerms=385">Encyclopedia of Combinatorial Structures 385</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F Self-convolution of rooted trees A000081.
%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, x=0, n+1), x,n): seq (a(n), n=2..28); # _Alois P. Heinz_, Aug 21 2008
%t <<NumericalDifferentialEquationAnalysis`; btc = ButcherTreeCount[max = 30]; Flatten[ Table[ ListConvolve[t=Take[btc, n], t], {n, 1, max}]] (* From Jean-François Alcover, Nov 02 2011 *)
%o (Haskell)
%o a000106 n = a000106_list !! (n-2)
%o a000106_list = drop 2 $ conv a000081_list [] where
%o conv (v:vs) ws = (sum $ zipWith (*) ws' $ reverse ws') : conv vs ws'
%o where ws' = v : ws
%o -- _Reinhard Zumkeller_, Jun 17 2013
%Y Cf. A000081, A000242, A000300, A000343, A000395.
%K nonn,nice,easy,changed
%O 2,2
%A _N. J. A. Sloane_.
%E More terms from _Christian G. Bower_, Nov 15 1999.
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