A326525 Sum of the eighth largest parts in the partitions of n into 9 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 18, 24, 28, 39, 46, 60, 69, 90, 105, 133, 149, 189, 216, 264, 297, 364, 412, 494, 553, 661, 743, 877, 972, 1149, 1280, 1493, 1650, 1922, 2126, 2454, 2702, 3107, 3429, 3916, 4291, 4895, 5374, 6086, 6647

Offset: 0

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326526, A326527, A326528, A326529, A326530, A326531, A326532.

Programs

Mathematica

Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,SquareFreeQ]&][[;;,8]]],{n,0,60}] (* Harvey P. Dale, Jan 30 2024 *)

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2 * p, where mu is the Möbius function (A008683).

a(n) = A326523(n) - A326524(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).