A326448 Sum of the fifth largest parts of the partitions of n into 8 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 10, 12, 18, 22, 32, 38, 52, 61, 82, 94, 124, 141, 180, 206, 262, 298, 373, 426, 527, 600, 735, 828, 1001, 1124, 1348, 1506, 1790, 1989, 2343, 2603, 3040, 3357, 3893, 4295, 4963, 5450, 6254, 6856, 7824, 8550, 9705

Offset

0,11

Links

Table of n, a(n) for n=0..54.

Index entries for sequences related to partitions

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 * l, where mu is the Möbius function (A008683).

a(n) = A326444(n) - A326445(n) - A326446(n) - A326447(n) - A326449(n) - A326450(n) - A326451(n) - A326452(n).

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[l * 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}]

Crossrefs

Cf. A008683, A326443, A326444, A326445, A326446, A326447, A326449, A326450, A326451, A326452.

Sequence in context: A359744 A308956 A309834 * A326528 A326633 A323587

Adjacent sequences: A326445 A326446 A326447 * A326449 A326450 A326451

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 06 2019

Status

approved