A121659 Number of partitions of n into parts with at most one part not greater than 2.

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 20, 25, 32, 40, 51, 63, 79, 97, 121, 148, 182, 221, 271, 328, 398, 479, 579, 694, 834, 995, 1190, 1415, 1684, 1995, 2366, 2793, 3298, 3881, 4569, 5360, 6288, 7355, 8603, 10037, 11705, 13619, 15842, 18388, 21333, 24703

Offset

1,4

Comments

The number of partitions of n that have the first part at least thrice larger than the second part. Example: a(8) = #{8, 7+1, 6+2, 6+1+1, 5+1+1+1, 4+1+1+1+1, 3+1+1+1+1+1}=7. [Mircea Merca, Jul 24 2011]

Links

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

Mircea Merca, Fast algorithm for generating ascending compositions, arXiv:1903.10797 [math.CO], 2019.

Formula

a(n) = A121081(n) - A008483(n-3) for n>2.

a(n) = p(n) - p(n-2) - p(n-3) + p(n-5) where p(n) = A000041(n). See Merca p. 6. - Michel Marcus, Mar 27 2019

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^2 / (4*sqrt(3)*n^2) * (1 - (18/Pi + 61*Pi/24)/sqrt(6*n)). - Vaclav Kotesovec, Jan 15 2022

Example

a(8) = #{8,7+1,6+2,5+3,4+4,4+3+1,3+3+2} = 7;
a(9) = #{9,8+1,7+2,6+3,5+4,5+3+1,4+4+1,4+3+2,3+3+3} = 9.

Crossrefs

Cf. A027336, A008483, A121081, A000041.

Sequence in context: A003413 A100853 A174065 * A096814 A039861 A039877

Adjacent sequences: A121656 A121657 A121658 * A121660 A121661 A121662

Keyword

nonn

Author

Reinhard Zumkeller, Aug 14 2006

Status

approved