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%I #15 Oct 07 2014 09:00:24
%S 1,2,5,14,42,131,420,1376,4589,15537,53293,184881,647752,2289215,
%T 8152147,29226618,105408688,382193502,1392377762,5094356032,
%U 18711122069,68965586862,255006331944,945662753514,3516281928159,13106923073426,48967229534156,183326438562982
%N Binomial transform of rooted tree numbers.
%H Alois P. Heinz, <a href="/A006930/b006930.txt">Table of n, a(n) for n = 0..600</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 1 + A051491 = 3.955765285651994974714..., c = 2.0132071688691451819... . - _Vaclav Kotesovec_, Aug 25 2014
%p with(numtheory):
%p b:= proc(n) option remember; local d, j; `if`(n<=1, n,
%p (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
%p end:
%p a:= n-> add(b(k+1)*binomial(n,k), k=0..n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 17 2013
%t b[n_] := b[n] = Module[{d, j}, If[n <= 1, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)]]; a[n_] := Sum[b[k+1]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Oct 07 2014, translated from Maple *)
%Y Cf. A000081, A051491.
%K nonn
%O 0,2
%A _N. J. A. Sloane_.
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