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Number of set partitions of [2n] in which the size of the first block is n.
5

%I #12 May 20 2018 11:36:49

%S 1,1,6,50,525,6552,93786,1504932,26640900,514083570,10713538550,

%T 239342496120,5697111804566,143759365731100,3829115870472600,

%U 107260549881604200,3149703964487098665,96686987797052290440,3094969650442399156350,103079905957566679518300

%N Number of set partitions of [2n] in which the size of the first block is n.

%C The blocks are ordered with increasing least elements.

%C a(0) = 1 by convention.

%H Alois P. Heinz, <a href="/A297926/b297926.txt">Table of n, a(n) for n = 0..445</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F a(n) = binomial(2*n-1,n-1) * Bell(n).

%F a(n) = A056857(2n,n) = A056860(2n,n).

%e a(1) = 1: 1|2.

%e a(2) = 6: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 14|2|3.

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

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

%p end:

%p a:= n-> binomial(2*n-1, n-1)*b(n):

%p seq(a(n), n=0..25);

%t b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j]*Binomial[n-1, j-1], {j, 1, n}]];

%t a[n_] := Binomial[2*n-1, n-1] * b[n];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 20 2018, translated from Maple *)

%Y Cf. A000110, A056857, A056860, A276961, A297924.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 08 2018