A308958 Sum of the third largest parts in the partitions of n into 7 squarefree parts.
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
Keywords
Programs
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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}]
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).