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A325481 Number of colored set partitions of [2n] where colors of the elements of subsets are distinct and in increasing order and exactly n colors are used.

%I A325481 #11 Dec 14 2020 08:32:08
%S A325481 1,1,41,8020,4396189,5226876501,11581358373398,43225961160925257,
%T A325481 252807246693691825421,2194141947654736889023357,
%U A325481 27084992620572948369385642201,459597167193175440533390098112664,10424556988338412210154331381461375830
%N A325481 Number of colored set partitions of [2n] where colors of the elements of subsets are distinct and in increasing order and exactly n colors are used.
%H A325481 Alois P. Heinz, <a href="/A325481/b325481.txt">Table of n, a(n) for n = 0..151</a>
%F A325481 a(n) = A322670(2n,n).
%p A325481 b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*
%p A325481       binomial(n-1, j-1)*binomial(k, j), j=1..min(k, n)))
%p A325481     end:
%p A325481 a:= n-> add(b(2*n, n-i)*(-1)^i*binomial(n, i), i=0..n):
%p A325481 seq(a(n), n=0..14);
%t A325481 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 A325481 a[n_] := Sum[b[2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
%t A325481 a /@ Range[0, 14] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A325481 Cf. A322670.
%K A325481 nonn
%O A325481 0,3
%A A325481 _Alois P. Heinz_, Sep 06 2019