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A145454 Exponential transform of binomial(n,4) = A000332.

%I A145454 #17 Nov 10 2016 03:49:46
%S A145454 1,0,0,0,1,5,15,35,105,756,6510,46530,283470,1667380,11457446,
%T A145454 99776040,969295145,9298091180,86154691680,804769174536,8052676029420,
%U A145454 88489327173660,1038440150703340,12501684521410700,151866259113256611
%N A145454 Exponential transform of binomial(n,4) = A000332.
%C A145454 a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 4 balls are seen at the top.
%C A145454 a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 4 labels.
%H A145454 Alois P. Heinz, <a href="/A145454/b145454.txt">Table of n, a(n) for n = 0..500</a>
%H A145454 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A145454 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A145454 E.g.f.: exp(exp(x)*x^4/4!).
%p A145454 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A145454       binomial(n-1, j-1) *binomial(j,4) *a(n-j), j=1..n))
%p A145454     end:
%p A145454 seq(a(n), n=0..30);
%t A145454 Table[Sum[BellY[n, k, Binomial[Range[n], 4]], {k, 0, n}], {n, 0, 25}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%Y A145454 4th column of A145460, A143398.
%K A145454 nonn
%O A145454 0,6
%A A145454 _Alois P. Heinz_, Oct 10 2008