A229816 Number of partitions of n such that if the length is k then k is not a part.

1, 0, 2, 2, 4, 5, 9, 11, 18, 23, 34, 44, 63, 80, 111, 142, 190, 242, 319, 402, 522, 655, 837, 1045, 1322, 1638, 2053, 2532, 3144, 3857, 4757, 5803, 7111, 8636, 10516, 12716, 15404, 18543, 22355, 26807, 32168, 38430, 45929, 54670, 65088, 77220, 91599, 108330, 128077, 151006, 177974

Offset: 0

Author

Jon Perry, Sep 30 2013

Keywords

• nonn

Comments

For example with n=5 neither 32 or 311 are allowed.

Conjecture: Also, for n>=1, a(n-1) is the total number of distinct parts of each partition of 2n with partition rank n. - George Beck, Jun 23 2019

Examples

			a(2) = 2 : 2, 11.
a(6) = 9 : 6, 51, 411, 33, 3111, 222, 2211, 21111, 111111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• 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. A116645.

Cf. A002865 (partitions where the number of parts is itself a part).

Cf. A000041, A002865.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(in, 0, b(n-i, i, t))))
    end:
a:= n-> b(n$2, 1)-b((n-1)$2, 2):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 30 2013

Mathematica

nn=50;CoefficientList[Series[ Product[1/(1-x^i),{i,1,nn}]-x Product[1/(1-x^i),{i,2,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 30 2013 *)
Table[PartitionsP[n] - (PartitionsP[n - 1] - PartitionsP[n - 2]), {n, 0, 60}] (* Vincenzo Librandi, Jun 30 2019 *)

PARI

N=66;  x='x+O('x^N);
gf = 1/eta(x) - x*(1-x)/eta(x);
Vec( gf )
\\ Joerg Arndt, Sep 30 2013

Formula

From Joerg Arndt, Sep 30 2013: (Start)

a(n) = A000041(n) - A002865(n-1), n>=1.

G.f.: 1/E(x) - x*(1-x)/E(x) where E(x) = Product_{k>=1} 1-x^k. (End)

Extensions

Corrected a(8) and extended beyond a(9), Joerg Arndt, Sep 30 2013