A326524 Sum of the smallest 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, 8, 9, 13, 17, 23, 26, 36, 42, 55, 63, 80, 93, 119, 131, 165, 188, 230, 255, 312, 351, 420, 466, 555, 620, 731, 804, 945, 1046, 1216, 1333, 1550, 1702, 1959, 2141, 2452, 2688, 3064, 3334, 3790, 4136, 4673, 5070

Offset: 0

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A326522, A326523, A326525, A326526, A326527, A326528, A326529, A326530, A326531, A326532.

Programs

Mathematica

Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,SquareFreeQ]&][[;;,-1]]],{n,0,60}] (* Harvey P. Dale, Mar 22 2023 *)

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

a(n) = A326523(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).