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A000395 6th power of rooted tree enumerator; number of linear forests of 6 rooted trees.

%I A000395 M4175 N1739 #37 Jan 03 2021 04:31:33
%S A000395 1,6,27,104,369,1236,3989,12522,38535,116808,350064,1039896,3068145,
%T A000395 9004182,26314773,76652582,222705603,645731148,1869303857,5404655358,
%U A000395 15611296146,45060069406,129989169909,374843799786,1080624405287
%N A000395 6th power of rooted tree enumerator; number of linear forests of 6 rooted trees.
%D A000395 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
%D A000395 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000395 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000395 Alois P. Heinz, <a href="/A000395/b000395.txt">Table of n, a(n) for n = 6..1000</a>
%H A000395 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A000395 G.f.: B(x)^6 where B(x) is g.f. of A000081.
%F A000395 a(n) ~ 6 * A187770 * A051491^n / n^(3/2). - _Vaclav Kotesovec_, Jan 03 2021
%p A000395 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-5)^6, x=0, n+1), x,n): seq(a(n), n=6..30);  # _Alois P. Heinz_, Aug 21 2008
%t A000395 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-5]^6, {x, 0, n}]; Table[a[n], {n, 6, 30}] (* _Jean-François Alcover_, Oct 13 2014, after _Alois P. Heinz_ *)
%Y A000395 Column 6 of A339067.
%Y A000395 Cf. A000081, A000106, A000242, A000300, A000343.
%K A000395 nonn
%O A000395 6,2
%A A000395 _N. J. A. Sloane_
%E A000395 More terms from _Christian G. Bower_, Nov 15 1999