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A297926 Number of set partitions of [2n] in which the size of the first block is n.

%I A297926 #12 May 20 2018 11:36:49
%S A297926 1,1,6,50,525,6552,93786,1504932,26640900,514083570,10713538550,
%T A297926 239342496120,5697111804566,143759365731100,3829115870472600,
%U A297926 107260549881604200,3149703964487098665,96686987797052290440,3094969650442399156350,103079905957566679518300
%N A297926 Number of set partitions of [2n] in which the size of the first block is n.
%C A297926 The blocks are ordered with increasing least elements.
%C A297926 a(0) = 1 by convention.
%H A297926 Alois P. Heinz, <a href="/A297926/b297926.txt">Table of n, a(n) for n = 0..445</a>
%H A297926 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A297926 a(n) = binomial(2*n-1,n-1) * Bell(n).
%F A297926 a(n) = A056857(2n,n) = A056860(2n,n).
%e A297926 a(1) = 1: 1|2.
%e A297926 a(2) = 6: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 14|2|3.
%p A297926 b:= proc(n) option remember; `if`(n=0, 1,
%p A297926       add(b(n-j)*binomial(n-1, j-1), j=1..n))
%p A297926     end:
%p A297926 a:= n-> binomial(2*n-1, n-1)*b(n):
%p A297926 seq(a(n), n=0..25);
%t A297926 b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j]*Binomial[n-1, j-1], {j, 1, n}]];
%t A297926 a[n_] := Binomial[2*n-1, n-1] * b[n];
%t A297926 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 20 2018, translated from Maple *)
%Y A297926 Cf. A000110, A056857, A056860, A276961, A297924.
%K A297926 nonn
%O A297926 0,3
%A A297926 _Alois P. Heinz_, Jan 08 2018