A162932 a(n) = A053445(n-2) - A053445(n-4).

1, 0, 1, 1, 1, 1, 3, 1, 3, 4, 4, 4, 8, 6, 10, 11, 13, 15, 22, 20, 28, 33, 39, 43, 58, 60, 77, 88, 104, 119, 148, 160, 197, 226, 265, 300, 363, 404, 481, 549, 638, 727, 858, 961, 1126, 1283, 1480, 1680, 1953, 2201, 2544, 2887, 3309, 3750, 4312, 4857, 5566, 6301, 7175

Offset

6,7

Comments

a(n) counts partitions of n such that all parts are >=2 and the largest part occurs at least three times, see example.

Links

Andrew van den Hoeven, Table of n, a(n) for n = 6..10000

Formula

From Mircea Merca, Jun 11 2012: (Start)

a(n) = p(n) - 2*p(n-1) + 2*p(n-3) - p(n-4) for n >= 6, where p(n) = A000041(n).

G.f.: -1 + x - x^3 + (1 - x)*Product_{k > 2} 1/(1 - x^k). (End)

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (36*sqrt(2)*n^(5/2)). - Vaclav Kotesovec, Jun 02 2018

G.f.: Sum_{n >= 1} q^(3*n+3)/Product_{k = 1..n} 1 - q^(k+1). - Peter Bala, Dec 01 2024

Example

For n = 19 the a(19) = 6 partitions are 5554, 44443, 55522, 444322, 3333322 and 33322222.

Mathematica

Table[PartitionsP[n] - 2 PartitionsP[n - 1] + 2 PartitionsP[n - 3] - PartitionsP[n - 4], {n, 6, 70}] (* Vincenzo Librandi, Dec 09 2014 *)

Prog

(Magma) a:=func<n | NumberOfPartitions(n)-2*NumberOfPartitions(n-1)+2*NumberOfPartitions(n-3)-NumberOfPartitions(n-4)>; [a(n): n in [6..100]]; // Vincenzo Librandi, Dec 09 2014

Crossrefs

Cf. A162931, A079946.

Sequence in context: A320279 A175290 A089958 * A216036 A008924 A175186

Adjacent sequences: A162929 A162930 A162931 * A162933 A162934 A162935

Keyword

nonn,changed

Author

Alford Arnold, Jul 17 2009

Extensions

Keyword:tabf removed, indexing corrected, sequence extended by R. J. Mathar, Sep 17 2009

Status

approved