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A305198 Number of set partitions of [2n+1] with symmetric block size list of length A109613(n).

%I A305198 #9 Dec 08 2020 08:04:11
%S A305198 1,1,7,56,470,10299,91925,3939653,36298007,2571177913,24158837489,
%T A305198 2557117944391,24350208829581,3601150175699409,34626777577615921,
%U A305198 6820331445080882282,66066554102006208712,16719951521837764142510,162903256982698962545956
%N A305198 Number of set partitions of [2n+1] with symmetric block size list of length A109613(n).
%H A305198 Alois P. Heinz, <a href="/A305198/b305198.txt">Table of n, a(n) for n = 0..200</a>
%F A305198 a(n) = A275281(2n+1,A109613(n)).
%p A305198 b:= proc(n, s) option remember; expand(`if`(n>s,
%p A305198       binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
%p A305198       b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
%p A305198     end:
%p A305198 a:= n-> coeff(b(2*n+1, 0), x, n+irem(n+1, 2)):
%p A305198 seq(a(n), n=0..20);
%t A305198 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 A305198 a[n_] := Coefficient[b[2n + 1, 0], x, n + Mod[n + 1, 2]];
%t A305198 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)
%Y A305198 Bisection (odd part) of A305197.
%Y A305198 Cf. A109613, A275281, A275283.
%K A305198 nonn
%O A305198 0,3
%A A305198 _Alois P. Heinz_, May 27 2018