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A326528
Sum of the fifth largest parts of the partitions of n into 9 squarefree parts.
10
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 10, 12, 18, 24, 34, 40, 56, 67, 91, 105, 138, 162, 209, 237, 304, 352, 441, 504, 630, 726, 893, 1016, 1236, 1409, 1700, 1912, 2287, 2579, 3052, 3417, 4018, 4492, 5237, 5824, 6756, 7508, 8655, 9561, 10967, 12114
OFFSET
0,12
FORMULA
a(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 * l, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326527(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).
MATHEMATICA
Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[l * 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