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A276961 Number of set partitions of [2n] with largest set of size n.

%I A276961 #31 Jan 18 2022 14:12:20
%S A276961 1,1,9,90,1015,12978,187110,3008148,53275365,1028142830,21426984722,
%T A276961 478684639524,11394222257054,287518726261900,7658231720886900,
%U A276961 214521099685649640,6299407928673657135,193373975592937777770,6189939300880260745050,206159811915115686404700
%N A276961 Number of set partitions of [2n] with largest set of size n.
%C A276961 The blocks are ordered with increasing least elements.
%C A276961 a(0) = 1 by convention.
%H A276961 Alois P. Heinz, <a href="/A276961/b276961.txt">Table of n, a(n) for n = 0..445</a>
%H A276961 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A276961 a(n) = A080510(2n,n).
%F A276961 a(n) = A327884(2n,n).
%F A276961 a(n) = ceiling(C(2n,n)*(A000110(n)-1/2)). - _Ludovic Schwob_, Jan 15 2022
%e A276961 a(1) = 1: 1|2.
%e A276961 a(2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
%p A276961 b:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A276961       b(n-i, k)*binomial(n-1, i-1), i=1..min(n, k)))
%p A276961     end:
%p A276961 a:= n-> `if`(n=0, 1, b(2*n, n)-b(2*n, n-1)):
%p A276961 seq(a(n), n=0..20);
%t A276961 b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - i, k]*Binomial[n - 1, i - 1], {i, 1, Min[n, k]}]];
%t A276961 a[n_] := If[n == 0, 1, b[2*n, n] - b[2*n, n - 1]];
%t A276961 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 20 2018, translated from Maple *)
%Y A276961 Cf. A000110, A080510, A276923, A297924, A297926, A327884, A328156.
%K A276961 nonn
%O A276961 0,3
%A A276961 _Alois P. Heinz_, Sep 22 2016