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A325482 Number of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and exactly two colors are used.

%I A325482 #15 Dec 15 2020 16:27:24
%S A325482 3,12,41,140,497,1848,7191,29184,123107,538076,2430353,11317644,
%T A325482 54229905,266906856,1347262319,6965034368,36833528195,199037675052,
%U A325482 1097912385849,6176578272780,35409316648433,206703355298072,1227820993510151,7416522514174080
%N A325482 Number of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and exactly two colors are used.
%H A325482 Alois P. Heinz, <a href="/A325482/b325482.txt">Table of n, a(n) for n = 2..792</a>
%F A325482 E.g.f.: 1-2*exp(x)+exp(x*(x+4)/2).
%F A325482 a(n) ~ n^(n/2) * exp(-1 + 2*sqrt(n) - n/2) / sqrt(2). - _Vaclav Kotesovec_, Sep 18 2019
%e A325482 a(3) = 12: 1a|2a3b, 1b|2a3b, 1a3b|2a, 1a3b|2b, 1a2b|3a, 1a2b|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
%p A325482 b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*
%p A325482       binomial(n-1, j-1)*binomial(k, j), j=1..min(k, n)))
%p A325482     end:
%p A325482 a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
%p A325482 seq(a(n), n=2..27);
%t A325482 b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k] Binomial[n - 1, j - 1] Binomial[k, j], {j, 1, Min[k, n]}]];
%t A325482 a[n_] := With[{k = 2}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
%t A325482 a /@ Range[2, 27] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A325482 Column k=2 of A322670.
%K A325482 nonn
%O A325482 2,1
%A A325482 _Alois P. Heinz_, Sep 06 2019