A308957 Sum of the fourth largest parts in the partitions of n into 7 squarefree parts.
0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 7, 11, 14, 20, 24, 36, 41, 56, 64, 86, 98, 129, 147, 193, 222, 284, 324, 409, 457, 567, 635, 773, 862, 1037, 1147, 1375, 1516, 1778, 1953, 2290, 2510, 2920, 3186, 3680, 4017, 4614, 4996, 5734, 6226, 7081, 7682, 8732, 9450
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[Sum[Sum[Sum[Sum[Sum[k * 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}] Table[Total[Select[IntegerPartitions[n,{7}],AllTrue[#,SquareFreeQ]&][[;;,4]]],{n,0,60}] (* Harvey P. Dale, Sep 24 2023 *)
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 * k, where mu is the Möbius function (A008683).