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A308958
Sum of the third largest parts in the partitions of n into 7 squarefree parts.
8
0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 14, 17, 25, 30, 44, 50, 72, 83, 115, 136, 184, 213, 278, 321, 409, 463, 579, 650, 807, 900, 1089, 1215, 1462, 1610, 1926, 2133, 2520, 2772, 3258, 3586, 4195, 4587, 5327, 5847, 6780, 7376, 8513, 9283, 10639, 11538, 13168
OFFSET
0,10
FORMULA
a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3} Sum_{i=j..floor((n-j-k-l-m-o)/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)^2 * j, where mu is the Möbius function (A008683).
a(n) = A308953(n) - A308954(n) - A308955(n) - A308956(n) - A308957(n) - A308959(n) - A308960(n).
MATHEMATICA
Table[Sum[Sum[Sum[Sum[Sum[Sum[j * 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]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 03 2019
STATUS
approved