A240855 Number of partitions p of n into distinct parts including the number of parts.

0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 4, 3, 5, 6, 8, 9, 10, 12, 16, 18, 22, 25, 29, 34, 41, 48, 55, 64, 74, 84, 98, 114, 130, 150, 170, 195, 222, 252, 287, 328, 371, 420, 475, 536, 604, 682, 766, 862, 970, 1088, 1218, 1365, 1526, 1704, 1904, 2124, 2366, 2637, 2934

Offset: 0

Author

Clark Kimberling, Apr 14 2014

Keywords

• nonn

Examples

			a(10) counts these 4 partitions:  82, 631, 532, 4321.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 101 terms from John Tyler Rascoe)
• 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, A008967, A240861.

Programs

Maple

h:= (p, i)-> add(coeff(p, x, j)*x^j, j=i+1..degree(p)):
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 [h(g[1], i), g[2]])(b(n, i-1, p)+
     (f-> f+[0, coeff(f[1], x, i)])(b(n-i, min(n-i, i-1), p+1)))))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..58);  # 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}] (* this sequence *)
Table[Count[f[n], p_ /; !MemberQ[p, Length[p]]], {n, 0, z}] (* A240861 *)

PARI

p_q(k) = {prod(j=1,k, 1-q^j);}
mGB_q(N,M) = {p_q(N+M)/(p_q(M)*(p_q(N)^2));}
A_q(N) = {my(q='q+O('q^N), g=sum(i=1, N, sum(j=1, i, q^((i*(i+1)/2)+(j*(j-1))) * mGB_q(j-1,i-j))));
concat([0], Vec(g)) }
A_q(50) \\ John Tyler Rascoe, Mar 13 2024

Formula

a(n) = A000009(n) - A240861(n).

G.f.: Sum_{i>0} Sum_{j=1..i} q^((i*(i+1)/2) + j*(j-1)) * [j-1,i-j]q, where [N,M]_q = Product{j=1..N+M}(1-q^j) / (Product_{j=1..M}(1-q^j) * (Product_{j=1..N}(1-q^j))^2). - John Tyler Rascoe, Mar 13 2024