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%I #6 Jul 12 2019 07:18:27
%S 0,0,0,0,0,0,0,0,0,9,10,22,24,52,70,120,144,221,288,418,500,714,858,
%T 1173,1392,1850,2184,2862,3304,4263,4950,6231,7136,8910,10234,12530,
%U 14256,17316,19684,23673,26640,31816,35826,42355,47388,55755,62284,72662
%N Sum of all the parts in the partitions of n into 9 squarefree parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = n * Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^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-q)^2, where mu is the Möbius function (A008683).
%F a(n) = n * A326522(n).
%F a(n) = A326524(n) + A326525(n) + A326526(n) + A326527(n) + A326528(n) + A326529(n) + A326530(n) + A326531(n) + A326532(n).
%t Table[n*Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[MoebiusMu[q]^2 * MoebiusMu[p]^2 * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l - m - o - p - q]^2, {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 80}]
%Y Cf. A008683, A326522, A326524, A326525, A326526, A326527, A326528, A326529, A326530, A326531, A326532.
%K nonn
%O 0,10
%A _Wesley Ivan Hurt_, Jul 11 2019
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