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A324523 Number of colored set partitions of [2n] where elements of subsets have distinct colors and exactly n colors are used.

%I A324523 #33 May 05 2020 05:33:21
%S A324523 1,1,74,31770,42687960,134092967400,831428629796160,
%T A324523 9095459029214397840,162061482211484681105280,
%U A324523 4429476877635332233622271360,177245727799376537644530489120000,10002691163041098923871227379695673600,772102922309973700712743861257373871078400
%N A324523 Number of colored set partitions of [2n] where elements of subsets have distinct colors and exactly n colors are used.
%H A324523 Alois P. Heinz, <a href="/A324523/b324523.txt">Table of n, a(n) for n = 0..139</a>
%H A324523 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A324523 a(n) = A323128(2n,n).
%p A324523 b:= proc(n, k) option remember; `if`(n=0, 1, add(k!/(k-j)!
%p A324523       *binomial(n-1, j-1)*b(n-j, k), j=1..min(k, n)))
%p A324523     end:
%p A324523 a:= n-> add(b(2*n, n-i)*(-1)^i*binomial(n, i), i=0..n):
%p A324523 seq(a(n), n=0..15);
%t A324523 b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[k!/(k-j)! Binomial[n - 1, j - 1]* b[n - j, k], {j, 1, Min[k, n]}]];
%t A324523 a[n_] := Sum[b[2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
%t A324523 a /@ Range[0, 15] (* _Jean-François Alcover_, May 05 2020, after Maple *)
%Y A324523 Cf. A323128.
%K A324523 nonn
%O A324523 0,3
%A A324523 _Alois P. Heinz_, Sep 03 2019