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%I M3479 N1414 #35 Jan 03 2021 04:30:43
%S 1,4,14,44,133,388,1116,3168,8938,25100,70334,196824,550656,1540832,
%T 4314190,12089368,33911543,95228760,267727154,753579420,2123637318,
%U 5991571428,16923929406,47857425416,135478757308,383929643780,1089118243128,3092612497260
%N 4th power of rooted tree enumerator: linear forests of 4 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 Alois P. Heinz, <a href="/A000300/b000300.txt">Table of n, a(n) for n = 4..500</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F G.f.: B(x)^4 where B(x) is g.f. of A000081.
%F a(n) ~ 4 * A187770 * A051491^n / n^(3/2). - _Vaclav Kotesovec_, Jan 03 2021
%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-3)^4, x=0, n+1), x,n): seq(a(n), n=4..30); # _Alois P. Heinz_, Aug 21 2008
%t 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, n/k}]; bb[n_] := bb[n] = Sum[b[k]*x^k, {k, 1, n}]; a[n_] := Coefficient[ Series[ bb[n - 3]^4, {x, 0, n + 1}], x, n]; Table[a[n], {n, 4, 31}] (* _Jean-François Alcover_, Jan 25 2013, translated from _Alois P. Heinz_'s Maple program *)
%Y Column 4 of A339067.
%Y Cf. A000081, A000106, A000242, A000343, A000395.
%K nonn
%O 4,2
%A _N. J. A. Sloane_
%E More terms from _Christian G. Bower_, Nov 15 1999
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