A308909 Sum of the third largest parts in the partitions of n into 6 squarefree parts.

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 14, 15, 23, 28, 39, 43, 62, 70, 98, 115, 152, 175, 227, 253, 319, 356, 441, 485, 599, 656, 793, 864, 1026, 1121, 1344, 1453, 1709, 1865, 2184, 2357, 2747, 2964, 3449, 3719, 4289, 4618, 5330, 5693, 6494, 6956, 7922, 8430

Offset: 0

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A308902, A308903, A308906, A308907, A308908, A308910, A308911.

Programs

Mathematica

Table[Total[Select[IntegerPartitions[n,{6}],AllTrue[#,SquareFreeQ]&][[All,3]]],{n,0,60}] (* Harvey P. Dale, Jan 31 2022 *)

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-k-j-l-m)^2 * j, where mu is the Möbius function (A008683).

a(n) = A308903(n) - A308906(n) - A308907(n) - A308908(n) - A308910(n) - A308911(n).