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A308953
Sum of all the parts in the partitions of n into 7 squarefree parts.
8
0, 0, 0, 0, 0, 0, 0, 7, 8, 18, 20, 44, 60, 104, 126, 180, 224, 340, 396, 551, 640, 882, 1034, 1357, 1536, 2025, 2314, 2943, 3304, 4176, 4680, 5797, 6464, 7887, 8806, 10605, 11664, 14023, 15504, 18291, 20040, 23657, 25956, 30272, 32956, 38295, 41860, 48269
OFFSET
0,8
FORMULA
a(n) = 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, where mu is the Möbius function (A008683).
a(n) = n * A308952(n).
a(n) = A308954(n) + A308955(n) + A308956(n) + A308957(n) + A308958(n) + A308959(n) + A308960(n).
MATHEMATICA
Table[Total[Flatten[Select[IntegerPartitions[n, {7}], AllTrue[#, SquareFreeQ]&]]], {n, 0, 50}] (* Harvey P. Dale, Feb 25 2024 *)
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 03 2019
STATUS
approved