A326527 Sum of the sixth largest parts of the partitions of n into 9 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 9, 11, 16, 21, 29, 35, 49, 57, 77, 91, 118, 137, 177, 202, 255, 293, 363, 413, 509, 580, 707, 802, 969, 1097, 1319, 1481, 1764, 1980, 2337, 2615, 3069, 3421, 3982, 4431, 5126, 5689, 6553, 7240, 8301, 9169, 10451

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, A326525, A326526, A326528, A326529, A326530, A326531, A326532.

Programs

Mathematica

Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,SquareFreeQ]&][[All,6]]],{n,0,60}] (* Harvey P. Dale, Jul 05 2022 *)

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 * m, where mu is the Möbius function (A008683).

a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326528(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).