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A328153 Number of set partitions of [n] such that at least one of the block sizes is 3.

%I A328153 #13 May 02 2020 04:50:22
%S A328153 0,0,0,1,4,20,90,455,2352,13132,76540,473660,3069220,20922330,
%T A328153 149021600,1109629885,8604815520,69437698160,581661169640,
%U A328153 5051885815603,45411759404560,421977921782270,4047693372023070,40034523497947132,407818256494533984,4274309903558446900
%N A328153 Number of set partitions of [n] such that at least one of the block sizes is 3.
%H A328153 Alois P. Heinz, <a href="/A328153/b328153.txt">Table of n, a(n) for n = 0..576</a>
%H A328153 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A328153 E.g.f.: exp(exp(x)-1) - exp(exp(x)-1-x^3/6).
%F A328153 a(n) = A000110(n) - A124504(n).
%p A328153 b:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A328153       `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))
%p A328153     end:
%p A328153 a:= n-> b(n, 0)-b(n, 3):
%p A328153 seq(a(n), n=0..27);
%t A328153 b[n_, k_] := b[n, k] = If[n==0, 1, Sum[If[j==k, 0, b[n-j, k] Binomial[n-1, j-1]], {j, 1, n}]];
%t A328153 a[n_] := b[n, 0] - b[n, 3];
%t A328153 a /@ Range[0, 27] (* _Jean-François Alcover_, May 02 2020, after Maple *)
%Y A328153 Column k=3 of A327884.
%Y A328153 Cf. A000110, A124504, A328155.
%K A328153 nonn
%O A328153 0,5
%A A328153 _Alois P. Heinz_, Oct 05 2019