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

Author

Seiichi Manyama, May 22 2023

Keywords

• nonn

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

Examples

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

Links

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

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.

Programs

Mathematica

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

PARI

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

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