A326630 Sum of the eighth largest parts in the partitions of n into 10 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 18, 26, 30, 41, 50, 66, 77, 100, 117, 152, 174, 219, 252, 314, 357, 436, 499, 605, 685, 820, 929, 1109, 1243, 1469, 1650, 1947, 2169, 2536, 2833, 3297, 3663, 4235, 4707, 5424, 6000, 6867, 7604, 8684

Offset: 0

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A005117, A326626, A326627, A326628, A326629, A326631, A326632, A326633, A326634, A326635, A326636, A326637.

Programs

Mathematica

Table[Select[IntegerPartitions[n,{10}],AllTrue[#,SquareFreeQ]&][[All,8]]//Total,{n,0,60}] (* Harvey P. Dale, Apr 19 2020 *)

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 * p, where mu is the Möbius function (A008683).

a(n) = A326627(n) - A326628(n) - A326629(n) - A326631(n) - A326632(n) - A326633(n) - A326634(n) - A326635(n) - A326636(n) - A326637(n).