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A326523
Sum of all the parts in the partitions of n into 9 squarefree parts.
10
0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 10, 22, 24, 52, 70, 120, 144, 221, 288, 418, 500, 714, 858, 1173, 1392, 1850, 2184, 2862, 3304, 4263, 4950, 6231, 7136, 8910, 10234, 12530, 14256, 17316, 19684, 23673, 26640, 31816, 35826, 42355, 47388, 55755, 62284, 72662
OFFSET
0,10
FORMULA
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).
a(n) = n * A326522(n).
a(n) = A326524(n) + A326525(n) + A326526(n) + A326527(n) + A326528(n) + A326529(n) + A326530(n) + A326531(n) + A326532(n).
MATHEMATICA
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}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 11 2019
STATUS
approved