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

0,7

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.

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

Example

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

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<n, [0$2],
     `if`(n=0, [x^p, 0], (g-> [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 *)

Prog

(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

Crossrefs

Cf. A000009, A008967, A240861.

Sequence in context: A162330 A134967 A320581 * A084612 A295164 A241419

Adjacent sequences: A240852 A240853 A240854 * A240856 A240857 A240858

Keyword

nonn

Author

Clark Kimberling, Apr 14 2014

Status

approved