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A324241 Number of set partitions of [2n] where each subset is again partitioned into n nonempty subsets.

%I A324241 #30 May 08 2022 08:24:39
%S A324241 1,2,10,100,1736,42651,1324114,49330996,2141770488,106175420065,
%T A324241 5917585057033,366282501223002,24930204592110338,1850568574258750360,
%U A324241 148782988064395367700,12879868072770703598760,1194461517469808134322280,118144018577011379763287565
%N A324241 Number of set partitions of [2n] where each subset is again partitioned into n nonempty subsets.
%H A324241 Alois P. Heinz, <a href="/A324241/b324241.txt">Table of n, a(n) for n = 0..345</a>
%H A324241 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A324241 a(n) = A324162(2n,n).
%F A324241 a(n) = A007820(n) + A088218(n) for n > 0. - _Seiichi Manyama_, May 08 2022
%e A324241 a(2) = 10: 123/4, 124/3, 12/34, 134/2, 13/24, 14/23, 1/234, 1/2|3/4, 1/3|2/4, 1/4|2/3.
%p A324241 b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)
%p A324241       *binomial(n-1, j-1)*Stirling2(j, k), j=k..n))
%p A324241     end:
%p A324241 a:= n-> b(2*n, n):
%p A324241 seq(a(n), n=0..18);
%t A324241 b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 0 || k > n, 0, Sum[b[n-j, k]* Binomial[n - 1, j - 1] StirlingS2[j, k], {j, k, n}]]];
%t A324241 a[n_] := b[2n, n];
%t A324241 a /@ Range[0, 18] (* _Jean-François Alcover_, May 05 2020, after Maple *)
%o A324241 (PARI) a(n) = if(n==0, 1, stirling(2*n, n, 2)+binomial(2*n, n)/2); \\ _Seiichi Manyama_, May 08 2022
%Y A324241 Cf. A007820, A088218, A324162.
%K A324241 nonn
%O A324241 0,2
%A A324241 _Alois P. Heinz_, Sep 02 2019