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%I #16 Jan 07 2022 11:10:22
%S 0,0,0,0,0,0,0,0,8,9,20,29,56,83,138,193,299,408,594,789,1115,1462,
%T 1977,2551,3382,4279,5550,6948,8856,10970,13742,16841,20832,25303,
%U 30892,37180,44972,53652,64276,76108,90424,106352,125353,146501,171544,199318
%N Sum of the squarefree parts of the partitions of n into 8 parts.
%H David A. Corneth, <a href="/A309484/b309484.txt">Table of n, a(n) for n = 0..9999</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F 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).
%t Table[Total[Select[Flatten[IntegerPartitions[n,{8}]],SquareFreeQ]],{n,0,50}] (* _Harvey P. Dale_, Jun 10 2021 *)
%Y Cf. A008683, A309461, A309478, A309479, A309480, A309481, A309482, A309484, A309485, A309486.
%K nonn
%O 0,9
%A _Wesley Ivan Hurt_, Aug 04 2019
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