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%I #11 Sep 17 2019 02:56:36
%S 0,0,0,0,0,1,2,5,5,13,19,32,35,52,66,96,104,147,177,240,263,356,392,
%T 514,543,712,763,972,1015,1271,1370,1652,1728,2084,2223,2623,2750,
%U 3275,3480,4087,4276,5011,5312,6141,6388,7443,7842,8987,9405,10753,11335
%N Sum of the largest parts in the partitions of n into 5 squarefree parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l)^2 * (n-i-j-k-l), where mu is the Möbius function (A008683).
%F a(n) = A308839(n) - A308841(n) - A308842(n) - A308843(n) - A308844(n).
%e The partitions of n into 5 parts for n = 10, 11, ..
%e 1+1+1+1+10
%e 1+1+1+2+9
%e 1+1+1+3+8
%e 1+1+1+4+7
%e 1+1+1+5+6
%e 1+1+1+1+9 1+1+2+2+8
%e 1+1+1+2+8 1+1+2+3+7
%e 1+1+1+3+7 1+1+2+4+6
%e 1+1+1+4+6 1+1+2+5+5
%e 1+1+1+5+5 1+1+3+3+6
%e 1+1+1+1+8 1+1+2+2+7 1+1+3+4+5
%e 1+1+1+2+7 1+1+2+3+6 1+1+4+4+4
%e 1+1+1+3+6 1+1+2+4+5 1+2+2+2+7
%e 1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6
%e 1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5
%e 1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5
%e 1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4
%e 1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4
%e 1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6
%e 1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5
%e 1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4
%e 1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4
%e 2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3
%e --------------------------------------------------------------------------
%e n | 10 11 12 13 14 ...
%e --------------------------------------------------------------------------
%e a(n) | 19 32 35 52 66 ...
%e --------------------------------------------------------------------------
%e - _Wesley Ivan Hurt_, Sep 16 2019
%t Table[Sum[Sum[Sum[Sum[(n - i - j - k - l) * MoebiusMu[l]^2*MoebiusMu[k]^2*MoebiusMu[j]^2*MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l]^2, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 80}]
%Y Cf. A008683, A308839, A308840, A308841, A308842, A308843, A308844.
%K nonn
%O 0,7
%A _Wesley Ivan Hurt_, Jun 28 2019