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A326636
Sum of the second largest parts of the partitions of n into 10 squarefree parts.
11
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 18, 22, 36, 45, 72, 88, 127, 153, 215, 263, 351, 418, 555, 658, 843, 984, 1252, 1460, 1825, 2118, 2623, 3029, 3697, 4248, 5168, 5914, 7101, 8088, 9676, 10960, 12974, 14647, 17246, 19396, 22653, 25384, 29527
OFFSET
0,13
FORMULA
a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} mu(r)^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-r)^2 * i, where mu is the Möbius function (A008683).
a(n) = A326627(n) - A326628(n) - A326629(n) - A326630(n) - A326631(n) - A326632(n) - A326633(n) - A326634(n) - A326635(n) - A326637(n).
MATHEMATICA
Table[Total[Select[IntegerPartitions[n, {10}], AllTrue[#, SquareFreeQ]&][[All, 2]]], {n, 0, 55}] (* Harvey P. Dale, Jan 03 2023 *)
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 14 2019
STATUS
approved