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

0,5

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.

Formula

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

Example

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

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<n, 0,
     `if`(n=0, [x^p, 0], (f-> [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 *)

Crossrefs

Cf. A000009, A240855.

Sequence in context: A282562 A035682 A054543 * A029046 A035372 A035576

Adjacent sequences: A240858 A240859 A240860 * A240862 A240863 A240864

Keyword

nonn,easy

Author

Clark Kimberling, Apr 14 2014

Extensions

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

Status

approved