A275283 Number of set partitions of [2n] with symmetric block size list of length n.

1, 1, 3, 19, 171, 2066, 31346, 559987, 11954993, 282835456, 7785919355, 229359684137, 7731656573016, 272633076900991, 10876116332074739, 446659746000614675, 20580725671071449149, 964732749192326683508, 50418595763262446272127, 2656265906893413392905767

Offset: 0

Author

Alois P. Heinz, Jul 21 2016

Keywords

• nonn

Examples

			a(0) = 1: {}.
a(1) = 1: 12.
a(2) = 3: 12|34, 13|24, 14|23.
a(3) = 19: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 13|25|46, 13|26|45, 14|23|56, 1|2345|6, 1|2346|5, 15|23|46, 1|2356|4, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 1|2456|3, 16|24|35, 15|26|34, 16|25|34.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Wikipedia, Partition of a set

Crossrefs

Bisection (even part) of A305197.

Cf. A275281.

Programs

Mathematica

b[n_, s_] := b[n, s] = Expand[If[n>s, Binomial[n-1, n-s-1]*x, 1] + Sum[Binomial[n-1, j-1]*b[n-j, s+j]*Binomial[s+j-1, j-1], {j, 1, (n-s)/2}]*x^2];
T[n_] := T[n] = Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
a[n_] := T[2n][[n+1]];
a /@ Range[0, 20] (* Jean-François Alcover, Aug 21 2021,after Alois P. Heinz in A275281 *)

Formula

a(n) = A275281(2n,n).

a(n) ~ c * n^(n-1/2) * d^n / (exp(n) * 2^(n-3/2)), where d = 5.99720652866734051428..., c = 0.331364442872654716... if n is even and c = 0.32118925729236323... if n is odd. - Vaclav Kotesovec, Aug 08 2016