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A000242 3rd power of rooted tree enumerator; number of linear forests of 3 rooted trees.

%I A000242 M2798 N1126 #37 Jan 03 2021 04:30:19
%S A000242 1,3,9,25,69,186,503,1353,3651,9865,26748,72729,198447,543159,1491402,
%T A000242 4107152,11342826,31408719,87189987,242603970,676524372,1890436117,
%U A000242 5292722721,14845095153,41708679697,117372283086,330795842217
%N A000242 3rd power of rooted tree enumerator; number of linear forests of 3 rooted trees.
%D A000242 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
%D A000242 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000242 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000242 T. D. Noe, <a href="/A000242/b000242.txt">Table of n, a(n) for n=3..200</a>
%H A000242 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A000242 G.f.: B(x)^3 where B(x) is g.f. of A000081.
%F A000242 a(n) ~ 3 * A187770 * A051491^n / n^(3/2). - _Vaclav Kotesovec_, Jan 03 2021
%p A000242 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-2)^3, x=0, n+1), x,n): seq(a(n), n=3..29); # _Alois P. Heinz_, Aug 21 2008
%t A000242 max = 29; 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]}]; f[x_] := Sum[ b[k]*x^k, {k, 0, max}]; Drop[ CoefficientList[ Series[f[x]^3, {x, 0, max}], x], 3] (* _Jean-François Alcover_, Oct 25 2011, after _Alois P. Heinz_ *)
%Y A000242 Column 3 of A339067.
%Y A000242 Cf. A000081, A000106, A000300, A000343, A000395.
%K A000242 nonn,easy,nice
%O A000242 3,2
%A A000242 _N. J. A. Sloane_
%E A000242 More terms from _Christian G. Bower_, Nov 15 1999