A363230 Number of partitions of n with rank 3 or higher (the rank of a partition is the largest part minus the number of parts).

0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, 70, 90, 120, 154, 201, 256, 330, 415, 529, 662, 833, 1035, 1293, 1595, 1976, 2425, 2982, 3640, 4449, 5401, 6565, 7935, 9592, 11543, 13891, 16645, 19943, 23808, 28408, 33792, 40172, 47619, 56413, 66661, 78708, 92724, 109149, 128213, 150486, 176293

Offset: 1

Author

Seiichi Manyama, May 22 2023

Keywords

• nonn

Examples

			a(6) = 2 counts these partitions: 6, 5+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), this sequence (r=3), A363231 (r=4).

Cf. A000041, A101200, A115067.

Programs

PARI

a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+5)/2));

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+5)/2).

a(n) = p(n-4) - p(n-11) + p(n-21) - ... + (-1)^(k-1) * p(n-k*(3*k+5)/2) + ..., where p() is A000041().

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 31*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023