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

0,3

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

Links

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

Formula

a(n) = A000041(n) - A002865(n-1), n>=1. [Joerg Arndt, Sep 30 2013]

G.f.: 1/E(x) - x*(1-x)/E(x) where E(x) = Product_{k>=1} 1-x^k. [Joerg Arndt, Sep 30 2013]

Example

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

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<t, 0,
       b(n, i-1, t)+`if`(i>n, 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, Juan 30 2019 *)

Prog

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

Crossrefs

Cf. A116645.

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

Cf. A000041, A002865.

Sequence in context: A326527 A326632 A240206 * A099574 A144118 A187069

Adjacent sequences: A229813 A229814 A229815 * A229817 A229818 A229819

Keyword

nonn

Author

Jon Perry, Sep 30 2013

Extensions

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

Status

approved