A326627 Sum of all the parts in the partitions of n into 10 squarefree parts.

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

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A326626, A326628, A326629, A326630, A326631, A326632, A326633, A326634, A326635, A326636, A326637.

Programs

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}]

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).