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A305197 Number of set partitions of [n] with symmetric block size list of length A004525(n).

%I A305197 #13 Jun 13 2018 14:11:15
%S A305197 1,1,1,1,3,7,19,56,171,470,2066,10299,31346,91925,559987,3939653,
%T A305197 11954993,36298007,282835456,2571177913,7785919355,24158837489,
%U A305197 229359684137,2557117944391,7731656573016,24350208829581,272633076900991,3601150175699409,10876116332074739
%N A305197 Number of set partitions of [n] with symmetric block size list of length A004525(n).
%H A305197 Alois P. Heinz, <a href="/A305197/b305197.txt">Table of n, a(n) for n = 0..400</a>
%F A305197 a(n) = A275281(n,(n+sin(n*Pi/2))/2).
%p A305197 b:= proc(n, s) option remember; expand(`if`(n>s,
%p A305197       binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
%p A305197       b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
%p A305197     end:
%p A305197 a:= n-> coeff(b(n, 0), x, (n+sin(n*Pi/2))/2):
%p A305197 seq(a(n), n=0..30);
%t A305197 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 A305197 a[n_] := Coefficient[b[n, 0], x, (n + Sin[n*Pi/2])/2];
%t A305197 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 13 2018, from Maple *)
%Y A305197 Bisections give A275283 (even part), A305198 (odd part).
%Y A305197 Cf. A004525, A275281.
%K A305197 nonn
%O A305197 0,5
%A A305197 _Alois P. Heinz_, May 27 2018