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

1, 1, 1, 1, 3, 7, 19, 56, 171, 470, 2066, 10299, 31346, 91925, 559987, 3939653, 11954993, 36298007, 282835456, 2571177913, 7785919355, 24158837489, 229359684137, 2557117944391, 7731656573016, 24350208829581, 272633076900991, 3601150175699409, 10876116332074739

Offset: 0

Author

Alois P. Heinz, May 27 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Crossrefs

Bisections give A275283 (even part), A305198 (odd part).

Cf. A004525, A275281.

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(n, 0), x, (n+sin(n*Pi/2))/2):
seq(a(n), n=0..30);

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[n, 0], x, (n + Sin[n*Pi/2])/2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 13 2018, from Maple *)

Formula

a(n) = A275281(n,(n+sin(n*Pi/2))/2).