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A327885 Number of set partitions of [n] such that at least one of the block sizes is 2.

%I A327885 #13 May 04 2020 07:04:37
%S A327885 0,0,1,3,9,35,150,672,3269,17271,97155,578985,3654750,24331320,
%T A327885 170074177,1244911605,9520843575,75890001665,629104453236,
%U A327885 5413637745144,48277814341765,445463898405225,4246785220234557,41775507558584283,423516880995944532
%N A327885 Number of set partitions of [n] such that at least one of the block sizes is 2.
%H A327885 Alois P. Heinz, <a href="/A327885/b327885.txt">Table of n, a(n) for n = 0..576</a>
%H A327885 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A327885 E.g.f.: exp(exp(x)-1) - exp(exp(x)-1-x^2/2).
%F A327885 a(n) = A000110(n) - A097514(n).
%e A327885 a(2) = 1: 12.
%e A327885 a(3) = 3: 12|3, 13|2, 1|23.
%e A327885 a(4) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
%e A327885 a(5) = 35: 123|45, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 145|23, 14|235, 14|23|5, 15|234, 15|23|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
%p A327885 b:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A327885       `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))
%p A327885     end:
%p A327885 a:= n-> b(n, 0)-b(n, 2):
%p A327885 seq(a(n), n=0..27);
%t A327885 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, n}]];
%t A327885 a[n_] := b[n, 0] - b[n, 2];
%t A327885 a /@ Range[0, 27] (* _Jean-François Alcover_, May 04 2020, after Maple *)
%Y A327885 Column k=2 of A327884.
%Y A327885 Cf. A000110, A097514.
%K A327885 nonn
%O A327885 0,4
%A A327885 _Alois P. Heinz_, Sep 28 2019