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

%I #16 Jun 13 2023 15:21:48

%S 1,1,8,122,2795,86472,3403127,164029595,9433737120,635182667816,

%T 49344452550230,4371727233798927,437489737355466560,

%U 49048715505983309703,6116937802946210183545,843220239072837883168510,127757559136845878072576947,21166434937698025552654090472

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

%C a(n) is the number of partitions of an 2n-set that contain at most n nonempty subsets.

%H Alois P. Heinz, <a href="/A359355/b359355.txt">Table of n, a(n) for n = 0..288</a>

%F a(n) = A102661(2n,n) for n >= 1. - _Alois P. Heinz_, Jun 13 2023

%p b:= proc(n) option remember; `if`(n=0, 1,

%p add(expand(b(n-j)*binomial(n-1, j-1)*x), j=1..n))

%p end:

%p a:= n-> (p-> add(coeff(p, x, i), i=0..n))(b(2*n, 0)):

%p seq(a(n), n=0..17); # _Alois P. Heinz_, Jun 13 2023

%o (PARI) a(n) = sum(j=0, n, stirling(2*n, j, 2)); \\ _Michel Marcus_, Dec 27 2022

%Y Cf. A359107, A048993, A102661.

%K nonn

%O 0,3

%A _Peter Luschny_, Dec 27 2022