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%I #9 Jan 07 2022 11:10:19
%S 0,0,0,0,0,0,0,0,0,0,10,11,24,35,66,97,160,223,343,485,714,952,1357,
%T 1808,2469,3237,4329,5576,7335,9322,12028,15148,19232,23934,30025,
%U 37070,45915,56180,68950,83661,101771,122540,147797,176821,211682,251515,299136
%N Sum of the squarefree parts of the partitions of n into 10 parts.
%H David A. Corneth, <a href="/A309486/b309486.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_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} (r * mu(r)^2 + q * mu(q)^2 + p * mu(p)^2 + o * mu(o)^2 + m * mu(m)^2 + l * mu(l)^2 + k * mu(k)^2 + j * mu(j)^2 + i * mu(i)^2 + (n-i-j-k-l-m-o-p-q-r) * mu(n-i-j-k-l-m-o-p-q-r)^2), where mu is the Möbius function (A008683).
%t Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(i * MoebiusMu[i]^2 + j * MoebiusMu[j]^2 + k * MoebiusMu[k]^2 + l * MoebiusMu[l]^2 + m * MoebiusMu[m]^2 + o * MoebiusMu[o]^2 + p * MoebiusMu[p]^2 + q * MoebiusMu[q]^2 + r * MoebiusMu[r]^2 + (n - i - j - k - l - m - o - p - q - r) * MoebiusMu[n - i - j - k - l - m - o - p - q - r]^2), {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 80}]
%Y Cf. A008683, A309464, A309478, A309479, A309480, A309481, A309482, A309484, A309485, A309486.
%K nonn
%O 0,11
%A _Wesley Ivan Hurt_, Aug 04 2019
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