A326447 Sum of the sixth largest parts in the partitions of n into 8 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 9, 11, 16, 19, 27, 33, 45, 51, 69, 80, 105, 117, 150, 172, 216, 242, 300, 339, 416, 466, 568, 636, 768, 852, 1022, 1135, 1348, 1483, 1748, 1934, 2260, 2481, 2876, 3163, 3655, 3993, 4582, 5014, 5735, 6244, 7098, 7732

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, A326448, A326449, A326450, A326451, A326452.

Programs

Mathematica

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

a(n) = A326444(n) - A326445(n) - A326446(n) - A326448(n) - A326449(n) - A326450(n) - A326451(n) - A326452(n).