A363231 Number of partitions of n with rank 4 or higher (the rank of a partition is the largest part minus the number of parts).

0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 72, 94, 124, 161, 209, 267, 343, 435, 551, 693, 870, 1084, 1351, 1672, 2066, 2542, 3121, 3815, 4658, 5664, 6875, 8319, 10049, 12102, 14553, 17452, 20894, 24959, 29766, 35420, 42089, 49911, 59100, 69856, 82452, 97152, 114324, 134315

Offset

1,7

Comments

In general, for r>=0, Sum_{k>=1} (-1)^(k-1) * p(n - k*(3*k + 2*r - 1)/2) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + (12*r-5)*Pi/144) / sqrt(n/6)), where p() is the partition function. - Vaclav Kotesovec, May 26 2023

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+7)/2).

a(n) = p(n-5) - p(n-13) + p(n-24) - ... + (-1)^(k-1) * p(n-k*(3*k+7)/2) + ..., where p() is A000041().

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 43*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023

Example

a(7) = 2 counts these partitions: 7, 6+1.

Mathematica

Table[Count[IntegerPartitions[n], _?(#[[1]]-Length[#]>3&)], {n, 60}] (* Harvey P. Dale, Jul 29 2024 *)

Prog

(PARI) a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+7)/2));

Crossrefs

With rank r or higher: A064174 (r=0), A064173 (r=1), A123975 (r=2), A363230 (r=3), this sequence (r=4).

Cf. A000041, A140090, A363213.

Sequence in context: A261775 A036007 A027342 * A184643 A307547 A182804

Adjacent sequences: A363228 A363229 A363230 * A363232 A363233 A363234

Keyword

nonn

Author

Seiichi Manyama, May 22 2023

Status

approved