A326526 Sum of the seventh 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, 8, 10, 15, 19, 26, 31, 43, 51, 67, 78, 103, 119, 152, 172, 219, 250, 308, 348, 429, 486, 585, 658, 794, 892, 1063, 1185, 1410, 1572, 1847, 2053, 2407, 2670, 3095, 3425, 3964, 4380, 5030, 5532, 6344, 6974, 7939

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, A326527, A326528, A326529, A326530, A326531, A326532.

Programs

Mathematica

Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,SquareFreeQ]&][[All,7]]],{n,0,60}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 05 2020 *)

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

a(n) = A326523(n) - A326524(n) - A326525(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).