A363434 Total number of blocks containing only elements of the same parity in all partitions of [n].

0, 1, 2, 7, 24, 97, 412, 1969, 9898, 54461, 313944, 1947613, 12603100, 86760255, 620559230, 4682462777, 36586620348, 299664171115, 2534306825064, 22355119509231, 203115201624030, 1917124624702475, 18598998656476220, 186822424157036439, 1925326063016510832

Offset: 0

Author

Alois P. Heinz, Jun 01 2023

Keywords

• nonn

Examples

			a(3) = 7 = 0 + 1 + 2 + 1 + 3 : 123, 12|3, 13|2, 1|23, 1|2|3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250
• Wikipedia, Partition of a set

Crossrefs

Cf. A000035, A124424, A363435, A363451, A363452, A363453.

Programs

Maple

b:= proc(n, e, o, m) option remember; `if`(n=0, e+o,
      (e+m)*b(n-1, o, e, m)+b(n-1, o, e+1, m)+
       `if`(o=0, 0, o*b(n-1, o-1, e, m+1)))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..24);

Mathematica

b[n_, e_, o_, m_] := b[n, e, o, m] = If[n == 0, e + o, (e + m)*b[n-1, o, e, m] + b[n - 1, o, e + 1, m] + If[o == 0, 0, o*b[n - 1, o - 1, e, m + 1]]];
a[n_] := b[n, 0, 0, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Sep 10 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} k * A124424(n,k).

a(n) = A363452(n) + A363453(n).

a(n) mod 2 = A000035(n).