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A326531
Sum of the second largest parts of the partitions of n into 9 squarefree parts.
10
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 18, 22, 36, 45, 70, 86, 124, 148, 207, 252, 334, 396, 520, 609, 781, 907, 1144, 1321, 1653, 1906, 2344, 2687, 3278, 3746, 4533, 5143, 6175, 6983, 8305, 9337, 11037, 12362, 14493, 16168, 18831, 20956, 24264, 26876
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 * i, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326532(n).
MATHEMATICA
Join[{0}, Table[Total[Select[IntegerPartitions[n, {9}], AllTrue[#, SquareFreeQ] &][[All, 2]]], {n, 60}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 08 2020 *)
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 11 2019
STATUS
approved