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A326627
Sum of all the parts in the partitions of n into 10 squarefree parts.
11
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 11, 24, 26, 56, 75, 128, 153, 234, 304, 460, 546, 770, 943, 1296, 1550, 2054, 2430, 3220, 3770, 4830, 5642, 7168, 8283, 10302, 11935, 14688, 16872, 20482, 23439, 28360, 32226, 38430, 43602, 51876, 58455, 68816, 77503, 90816
OFFSET
0,11
FORMULA
a(n) = 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)} mu(r)^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-r)^2, where mu is the Möbius function (A008683).
a(n) = n * A326626(n).
a(n) = A326628(n) + A326629(n) + A326630(n) + A326631(n) + A326632(n) + A326633(n) + A326634(n) + A326635(n) + A326636(n) + A326637(n).
MATHEMATICA
Table[n * Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[MoebiusMu[r]^2 * 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 - 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, 50}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 14 2019
STATUS
approved