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A309484
Sum of the squarefree parts of the partitions of n into 8 parts.
8
0, 0, 0, 0, 0, 0, 0, 0, 8, 9, 20, 29, 56, 83, 138, 193, 299, 408, 594, 789, 1115, 1462, 1977, 2551, 3382, 4279, 5550, 6948, 8856, 10970, 13742, 16841, 20832, 25303, 30892, 37180, 44972, 53652, 64276, 76108, 90424, 106352, 125353, 146501, 171544, 199318
OFFSET
0,9
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)} (i * mu(i)^2 + j * mu(j)^2 + k * mu(k)^2 + l * mu(l)^2 + m * mu(m)^2 + o * mu(o)^2 + p * mu(p)^2 + (n-i-j-k-l-m-o-p) * mu(n-i-j-k-l-m-o-p)^2), where mu is the Möbius function (A008683).
MATHEMATICA
Table[Total[Select[Flatten[IntegerPartitions[n, {8}]], SquareFreeQ]], {n, 0, 50}] (* Harvey P. Dale, Jun 10 2021 *)
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Aug 04 2019
STATUS
approved