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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.
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%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