A326450 Sum of the third largest parts of the partitions of n into 8 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 14, 17, 27, 32, 46, 55, 79, 93, 128, 153, 208, 245, 319, 375, 483, 556, 697, 799, 993, 1127, 1368, 1547, 1871, 2101, 2507, 2809, 3341, 3725, 4377, 4878, 5722, 6350, 7382, 8179, 9510, 10503, 12106, 13352, 15363, 16888

Offset: 0

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A326443, A326444, A326445, A326446, A326447, A326448, A326449, A326451, A326452.

Programs

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[j * 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) = 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 * j, where mu is the Möbius function (A008683).

a(n) = A326444(n) - A326445(n) - A326446(n) - A326447(n) - A326448(n) - A326449(n) - A326451(n) - A326452(n).