A186445 The number of partitions of n in which the first part is at least four times larger than the second part.

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 20, 25, 31, 38, 47, 57, 70, 85, 103, 124, 150, 180, 216, 258, 308, 366, 436, 516, 611, 721, 850, 1000, 1176, 1378, 1614, 1886, 2203, 2567, 2990, 3474, 4034, 4677, 5417, 6264, 7239, 8351, 9628

Offset

0,6

Links

Table of n, a(n) for n=0..50.

Formula

a(n) = p(n)-p(n-2)-p(n-3)-p(n-4)+p(n-5)+p(n-6)+p(n-7)-p(n-9), where p(n) = A000041(n).

a(n) ~ Pi^3 * exp(Pi*sqrt(2*n/3)) / (3*sqrt(2)*n^(5/2)) * (1 - (5*sqrt(6)/Pi + 109*Pi/(24*sqrt(6)))/sqrt(n)). - Vaclav Kotesovec, Jul 05 2025

Example

a(8) = #{8, 7+1, 6+1+1, 5+1+1+1, 4+1+1+1+1} = 5.
a(10) = #{10, 9+1, 8+2, 8+1+1, 7+1+1+1, 6+1+1+1+1, 5+1+1+1+1+1, 4+1+1+1+1+1+1} = 8.

Mathematica

Table[PartitionsP[n] - PartitionsP[n-2] - PartitionsP[n-3] - PartitionsP[n-4] + PartitionsP[n-5] + PartitionsP[n-6] + PartitionsP[n-7] - PartitionsP[n-9], {n, 0, 50}] (* Vaclav Kotesovec, Jul 05 2025 *)

Crossrefs

Cf. A000041.

Partial sums of A185325.

Sequence in context: A276828 A309715 A241089 * A080078 A036415 A054961

Adjacent sequences: A186442 A186443 A186444 * A186446 A186447 A186448

Keyword

nonn

Author

Mircea Merca, Feb 21 2011

Status

approved