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

%I A275283 #14 Aug 21 2021 06:33:48
%S A275283 1,1,3,19,171,2066,31346,559987,11954993,282835456,7785919355,
%T A275283 229359684137,7731656573016,272633076900991,10876116332074739,
%U A275283 446659746000614675,20580725671071449149,964732749192326683508,50418595763262446272127,2656265906893413392905767
%N A275283 Number of set partitions of [2n] with symmetric block size list of length n.
%H A275283 Alois P. Heinz, <a href="/A275283/b275283.txt">Table of n, a(n) for n = 0..200</a>
%H A275283 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A275283 a(n) = A275281(2n,n).
%F A275283 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
%e A275283 a(0) = 1: {}.
%e A275283 a(1) = 1: 12.
%e A275283 a(2) = 3: 12|34, 13|24, 14|23.
%e A275283 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.
%t A275283 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 A275283 T[n_] := T[n] = Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
%t A275283 a[n_] := T[2n][[n+1]];
%t A275283 a /@ Range[0, 20] (* _Jean-François Alcover_, Aug 21 2021,after _Alois P. Heinz_ in A275281 *)
%Y A275283 Bisection (even part) of A305197.
%Y A275283 Cf. A275281.
%K A275283 nonn
%O A275283 0,3
%A A275283 _Alois P. Heinz_, Jul 21 2016