%I #9 Jan 30 2024 22:24:05

%S 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,

%T 149,189,216,264,297,364,412,494,553,661,743,877,972,1149,1280,1493,

%U 1650,1922,2126,2454,2702,3107,3429,3916,4291,4895,5374,6086,6647

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F 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).

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

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

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

%K nonn

%O 0,12

%A _Wesley Ivan Hurt_, Jul 11 2019