A326444 Sum of all the parts in the partitions of n into 8 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 8, 9, 20, 22, 48, 65, 112, 135, 208, 255, 378, 456, 640, 756, 1034, 1219, 1632, 1875, 2444, 2835, 3640, 4147, 5220, 5952, 7392, 8382, 10234, 11550, 14004, 15688, 18810, 21021, 25040, 27798, 32802, 36421, 42680, 47160, 54832, 60489

Offset: 0

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A326443, A326445, A326446, A326447, A326448, A326449, A326450, A326451, A326452.

Programs

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[n * 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}]

Formula

a(n) = 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, where mu is the Möbius function (A008683).

a(n) = n * A326443(n).

a(n) = A326445(n) + A326446(n) + A326447(n) + A326448(n) + A326449(n) + A326450(n) + A326451(n) + A326452(n).