A363452 -id:A363452 - cp's OEIS frontend

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

Alois P. Heinz, Jun 01 2023

nonn

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

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

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

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);

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 *)

a(n) = Sum_{k=0..n} k * A124424(n,k).
a(n) = A363452(n) + A363453(n).
a(n) mod 2 = A000035(n).

A363453 Total number of blocks containing only even elements in all partitions of [n].

Original entry on oeis.org

0, 0, 1, 2, 12, 35, 206, 780, 4949, 22686, 156972, 837333, 6301550, 38122554, 310279615, 2090641920, 18293310174, 135445359397, 1267153412532, 10202944645270, 101557600812015, 881921432827544, 9299499328238110, 86508104545175503, 962663031508255416

Offset: 0

Alois P. Heinz, Jun 02 2023

nonn

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

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

Cf. A000110, A094577, A124422, A363434, A363452.

b:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
      add(Stirling2(i, k)*binomial(g, i)*
      add(Stirling2(u, j)*j^(g-i), j=0..u), i=k..g)
    end:
a:= n-> add(b(n, k)*k, k=0..floor(n/2)):
seq(a(n), n=0..25);
# second Maple program:
b:= proc(n, x, y, m) option remember; `if`(n=0, x,
      `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
      `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..25);

b[n_, x_, y_, m_] := b[n, x, y, m] = If[n == 0, x,
    If[x+m > 0, b[n-1, y, x, m]*(x+m), 0] + b[n-1, y, x+1, m] +
    If[y > 0, b[n-1, y-1, x, m+1]*y, 0]];
a[n_] := b[n, 0, 0, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 08 2023, after Alois P. Heinz *)

a(n) = Sum_{k=0..floor(n/2)} k * A124422(n,k).
a(n) = A363434(n) - A363452(n).
a(2n) = A363452(2n).
a(2n+1) = A363452(2n+1) - A094577(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula