A240861 Number of partitions p of n into distinct parts not including the number of parts.

1, 0, 1, 1, 2, 2, 2, 4, 4, 5, 6, 9, 10, 12, 14, 18, 22, 26, 30, 36, 42, 51, 60, 70, 81, 94, 110, 128, 148, 172, 198, 226, 260, 298, 342, 390, 446, 508, 577, 654, 742, 840, 951, 1074, 1212, 1366, 1538, 1728, 1940, 2176, 2440, 2732, 3056, 3416, 3814, 4254

Offset: 0

Author

Clark Kimberling, Apr 14 2014

Keywords

• nonn
• easy

Examples

			a(10) counts these 6 partitions:  {10}, {9,1}, {7,3}, {7,2,1}, {6,4}, {5,4,1}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4000
• Atul Dixit, Gaurav Kumar, and Aviral Srivastava, Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions, arXiv:2508.04359 [math.CO], 2025. See references.

Crossrefs

Cf. A000009, A240855.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 [add(coeff(f[1], x, j)*x^j
        , j=i+1..degree(f[1])), f[2]+coeff(f[1], x, i)])(
        b(n-i, min(n-i, i-1), p+1))+b(n, i-1, p)))
    end:
a:= n-> g(n)-b(n$2, 0)[2]:
seq(a(n), n=0..55);  # Alois P. Heinz, Mar 14 2024

Mathematica

z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; MemberQ[p, Length[p]]], {n, 0, z}]  (* A240855 *)
Table[Count[f[n], p_ /; !MemberQ[p, Length[p]]], {n, 0, z}] (* A240861 *)

Formula

a(n) = A000009(n) - A240855(n).

Extensions

a(0) changed to 1 by Alois P. Heinz, Mar 14 2024