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A326526
Sum of the seventh largest parts of the partitions of n into 9 squarefree parts.
10
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
0,12
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).
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 *)
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 11 2019
STATUS
approved