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A359355 a(n) = A359107(2*n, n) = Sum_{j=0..n} Stirling2(2*n, j) = Sum_{j=0..n} A048993(2*n, j).

%I A359355 #20 Jul 01 2025 09:03:41
%S A359355 1,1,8,122,2795,86472,3403127,164029595,9433737120,635182667816,
%T A359355 49344452550230,4371727233798927,437489737355466560,
%U A359355 49048715505983309703,6116937802946210183545,843220239072837883168510,127757559136845878072576947,21166434937698025552654090472
%N A359355 a(n) = A359107(2*n, n) = Sum_{j=0..n} Stirling2(2*n, j) = Sum_{j=0..n} A048993(2*n, j).
%C A359355 a(n) is the number of partitions of an 2n-set that contain at most n nonempty subsets.
%H A359355 Alois P. Heinz, <a href="/A359355/b359355.txt">Table of n, a(n) for n = 0..288</a>
%F A359355 a(n) = A102661(2n,n) for n >= 1. - _Alois P. Heinz_, Jun 13 2023
%p A359355 b:= proc(n) option remember; `if`(n=0, 1,
%p A359355       add(expand(b(n-j)*binomial(n-1, j-1)*x), j=1..n))
%p A359355     end:
%p A359355 a:= n-> (p-> add(coeff(p, x, i), i=0..n))(b(2*n, 0)):
%p A359355 seq(a(n), n=0..17);  # _Alois P. Heinz_, Jun 13 2023
%t A359355 b[n_] := b[n] = If[n == 0, 1, Sum[Expand[b[n-j]*Binomial[n-1, j-1]*x], {j, 1, n}]];
%t A359355 a[n_] := With[{p = b[2*n]}, Sum[Coefficient[p, x, i], {i, 0, n}]];
%t A359355 Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Jul 01 2025, after _Alois P. Heinz_ *)
%o A359355 (PARI) a(n) = sum(j=0, n, stirling(2*n, j, 2)); \\ _Michel Marcus_, Dec 27 2022
%Y A359355 Cf. A359107, A048993, A102661.
%K A359355 nonn
%O A359355 0,3
%A A359355 _Peter Luschny_, Dec 27 2022