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

0,4

Links

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

Formula

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

a(2n) = A282893(n).

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 *)

Crossrefs

Cf. A000009, A035363, A078408, A282893.

Sequence in context: A180876 A162170 A266770 * A008798 A005290 A326440

Adjacent sequences: A282889 A282890 A282891 * A282893 A282894 A282895

Keyword

nonn

Author

Robert G. Wilson v, Feb 24 2017

Status

approved