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Exponential transform of C(n,5) = A000389.
4

%I #14 Nov 21 2017 02:42:23

%S 1,0,0,0,0,1,6,21,56,126,378,3234,34056,289575,2020018,12237225,

%T 70634928,468041756,4190274648,45557515620,499503011496,5121432107757,

%U 49183015347774,462331794763069,4579813478536296,50959878972009546

%N Exponential transform of C(n,5) = A000389.

%C a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 5 balls are seen at the top.

%C a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 5 labels.

%H Alois P. Heinz, <a href="/A145455/b145455.txt">Table of n, a(n) for n = 0..500</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F E.g.f.: exp(exp(x)*x^5/5!).

%p a:= proc(n) option remember; `if`(n=0, 1,

%p add(binomial(n-1, j-1) *binomial(j,5) *a(n-j), j=1..n))

%p end:

%p seq(a(n), n=0..30);

%t With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x]*x^5/5!], {x, 0, nmax}], x]] (* _G. C. Greubel_, Nov 21 2017 *)

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(exp(x)*x^5/5!))) \\ _G. C. Greubel_, Nov 21 2017

%Y 5th column of A145460, A143398.

%K nonn

%O 0,7

%A _Alois P. Heinz_, Oct 10 2008