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

1,6

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+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

Example

a(6) = 2 counts these partitions: 6, 5+1.

Prog

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

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.

Sequence in context: A035995 A036006 A027341 * A262371 A359683 A184642

Adjacent sequences: A363227 A363228 A363229 * A363231 A363232 A363233

Keyword

nonn

Author

Seiichi Manyama, May 22 2023

Status

approved