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A325890 Number of colored set partitions of [n] where colors of the elements of subsets are in (weakly) increasing order and exactly two colors are used.

%I A325890 #11 Dec 15 2020 16:27:41
%S A325890 3,20,122,774,5247,38198,298139,2485690,22045130,207125874,2053771931,
%T A325890 21416863948,234145149539,2676207794512,31898152797430,
%U A325890 395584489687982,5093960430643323,67985187315217290,938835976835478467,13394336734762313862,197153821757472332126
%N A325890 Number of colored set partitions of [n] where colors of the elements of subsets are in (weakly) increasing order and exactly two colors are used.
%H A325890 Alois P. Heinz, <a href="/A325890/b325890.txt">Table of n, a(n) for n = 2..535</a>
%p A325890 b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*
%p A325890       binomial(n-1, j-1)*binomial(k+j-1, j), j=1..n))
%p A325890     end:
%p A325890 a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
%p A325890 seq(a(n), n=2..25);
%t A325890 b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k] Binomial[n - 1, j - 1] Binomial[k + j - 1, j], {j, 1, n}]];
%t A325890 a[n_] := With[{k = 2}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
%t A325890 a /@ Range[2, 25] (* _Jean-François Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)
%Y A325890 Column k=2 of A321296.
%K A325890 nonn
%O A325890 2,1
%A A325890 _Alois P. Heinz_, Sep 07 2019