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

1, 1, 7, 56, 470, 10299, 91925, 3939653, 36298007, 2571177913, 24158837489, 2557117944391, 24350208829581, 3601150175699409, 34626777577615921, 6820331445080882282, 66066554102006208712, 16719951521837764142510, 162903256982698962545956

Offset: 0

Author

Alois P. Heinz, May 27 2018

Keywords

• nonn

Links

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

Crossrefs

Bisection (odd part) of A305197.

Cf. A109613, A275281, A275283.

Programs

Maple

b:= proc(n, s) option remember; expand(`if`(n>s,
      binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
      b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
    end:
a:= n-> coeff(b(2*n+1, 0), x, n+irem(n+1, 2)):
seq(a(n), n=0..20);

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];
a[n_] := Coefficient[b[2n + 1, 0], x, n + Mod[n + 1, 2]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

Formula

a(n) = A275281(2n+1,A109613(n)).