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A326452
Sum of the largest parts of the partitions of n into 8 squarefree parts.
9
0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 5, 13, 19, 34, 40, 60, 80, 121, 147, 208, 253, 350, 417, 563, 651, 863, 1002, 1299, 1484, 1888, 2151, 2678, 3046, 3729, 4211, 5110, 5721, 6868, 7670, 9142, 10146, 11996, 13319, 15606, 17251, 20084, 22173, 25708, 28253, 32522
OFFSET
0,10
FORMULA
a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2 * (n-i-j-k-l-m-o-p), where mu is the Möbius function (A008683).
a(n) = A326444(n) - A326445(n) - A326446(n) - A326447(n) - A326448(n) - A326449(n) - A326450(n) - A326451(n).
MATHEMATICA
Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(n-i-j-k-l-m-o-p) * 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]^2, {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 06 2019
STATUS
approved