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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Wikipedia, Partition of a set

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

Example

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.

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 *)

Crossrefs

Bisection (even part) of A305197.

Cf. A275281.

Sequence in context: A143768 A256493 A353256 * A349768 A083071 A305459

Adjacent sequences: A275280 A275281 A275282 * A275284 A275285 A275286

Keyword

nonn

Author

Alois P. Heinz, Jul 21 2016

Status

approved