A326531 Sum of the second largest parts of the partitions of n into 9 squarefree parts.

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

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326525, A326526, A326527, A326528, A326529, A326530, A326532.

Programs

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 *)

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).