%I #11 Dec 15 2020 16:27:35
%S 1,2,122,30470,19946654,27291293442,67940872709600,280154891124993313,
%T 1787697422835498425966,16765591042116935170071062,
%U 221912878453525607344964295822,4012317533096874589918210188528948,96463460015261984561875523126569759208
%N Number of colored set partitions of [2n] where colors of the elements of subsets are in (weakly) increasing order and exactly n colors are used.
%H Alois P. Heinz, <a href="/A325889/b325889.txt">Table of n, a(n) for n = 0..150</a>
%F a(n) = A321296(2n,n).
%p b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*
%p binomial(n-1, j-1)*binomial(k+j-1, j), j=1..n))
%p end:
%p a:= n-> add(b(2*n, n-i)*(-1)^i*binomial(n, i), i=0..n):
%p seq(a(n), n=0..15);
%t 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 a[n_] := Sum[b[2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
%t a /@ Range[0, 15] (* _Jean-François Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)
%Y Cf. A321296.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Sep 07 2019