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A326443
Number of partitions of n into 8 squarefree parts.
11
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 15, 21, 24, 32, 36, 47, 53, 68, 75, 94, 105, 130, 143, 174, 192, 231, 254, 301, 330, 389, 424, 495, 539, 626, 678, 781, 847, 970, 1048, 1192, 1287, 1461, 1572, 1772, 1908, 2144, 2301, 2573, 2762, 3079, 3295
OFFSET
0,11
FORMULA
a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2, where mu is the Möbius function (A008683).
a(n) = A326444(n)/n for n > 0.
MATHEMATICA
Table[Count[IntegerPartitions[n, {8}], _?(AllTrue[#, SquareFreeQ]&)], {n, 0, 60}] (* Harvey P. Dale, Mar 10 2023 *)
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 06 2019
STATUS
approved