A282892 The difference between the number of partitions of n into odd parts (A000009) and the number of partitions of n into even parts (A035363).

0, 1, 0, 2, 0, 3, 1, 5, 1, 8, 3, 12, 4, 18, 7, 27, 10, 38, 16, 54, 22, 76, 33, 104, 45, 142, 64, 192, 87, 256, 120, 340, 159, 448, 215, 585, 283, 760, 374, 982, 486, 1260, 634, 1610, 814, 2048, 1049, 2590, 1335, 3264, 1700, 4097, 2146, 5120, 2708, 6378, 3390, 7917, 4243, 9792, 5276

Offset: 0

Author

Robert G. Wilson v, Feb 24 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..165 from Robert G. Wilson v)

Crossrefs

Cf. A000009, A035363, A078408, A282893.

Programs

Maple

with(numtheory):
b:= proc(n, t) option remember; `if`(n=0, 1, add(add(`if`(
      (d+t)::odd, d, 0), d=divisors(j))*b(n-j, t), j=1..n)/n)
    end:
a:= n-> b(n, 0) -b(n, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 24 2017

Mathematica

f[n_] := Length[ IntegerPartitions[n, All, 2Range[n] -1]] - Length[ IntegerPartitions[n, All, 2 Range[n]]]; Array[f, 60]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[Sum[If[
     OddQ[d+t], d, 0], {d, Divisors[j]}]*b[n-j, t], {j, 1, n}]/n];
a[n_] := b[n, 0] - b[n, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

Formula

a(2n-1) = A000009(2n-1) = A078408(n).

a(2n) = A282893(n).