A326636 Sum of the second largest parts of the partitions of n into 10 squarefree parts.

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

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A326626, A326627, A326628, A326629, A326630, A326631, A326632, A326633, A326634, A326635, A326637.

Programs

Mathematica

Table[Total[Select[IntegerPartitions[n,{10}],AllTrue[#,SquareFreeQ]&][[All,2]]],{n,0,55}] (* Harvey P. Dale, Jan 03 2023 *)

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).