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A145453 Exponential transform of binomial(n,3) = A000292(n-2).

%I A145453 #25 Sep 28 2017 07:46:12
%S A145453 1,0,0,1,4,10,30,175,1176,7084,42120,286605,2270180,19213766,
%T A145453 166326524,1497096055,14374680880,147259920760,1582837679056,
%U A145453 17659771122969,204674606377140,2473357218561250,31148510170120420,407154732691440811,5504706823227724904
%N A145453 Exponential transform of binomial(n,3) = A000292(n-2).
%C A145453 a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 3 balls are seen at the top.
%C A145453 a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 3 labels.
%H A145453 Seiichi Manyama, <a href="/A145453/b145453.txt">Table of n, a(n) for n = 0..530</a> (terms 0..200 from Alois P. Heinz)
%H A145453 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A145453 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A145453 E.g.f.: exp(exp(x)*x^3/3!).
%p A145453 a:= proc(n) option remember; `if`(n=0, 1,
%p A145453       add(binomial(n-1, j-1) *binomial(j,3) *a(n-j), j=1..n))
%p A145453     end:
%p A145453 seq(a(n), n=0..30);
%t A145453 Table[Sum[BellY[n, k, Binomial[Range[n], 3]], {k, 0, n}], {n, 0, 25}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%Y A145453 3rd column of A145460, A143398.
%Y A145453 Cf. A292889, A292910.
%K A145453 nonn
%O A145453 0,5
%A A145453 _Alois P. Heinz_, Oct 10 2008