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A326529
Sum of the fourth largest parts of the partitions of n into 9 squarefree parts.
10
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 7, 11, 14, 22, 28, 40, 47, 67, 80, 107, 125, 167, 197, 257, 298, 387, 455, 575, 663, 834, 964, 1185, 1353, 1648, 1882, 2263, 2553, 3045, 3438, 4054, 4542, 5331, 5965, 6936, 7716, 8936, 9936, 11434, 12646, 14504
OFFSET
0,12
FORMULA
a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/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)^2 * k, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326530(n) - A326531(n) - A326532(n).
MATHEMATICA
Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k * 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]^2, {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 80}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 11 2019
STATUS
approved