A308955 Sum of the sixth largest parts in the partitions of n into 7 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 9, 11, 14, 17, 25, 29, 39, 43, 58, 67, 85, 93, 120, 136, 168, 185, 229, 255, 311, 342, 414, 459, 547, 593, 711, 782, 911, 987, 1159, 1270, 1467, 1580, 1823, 1990, 2276, 2441, 2799, 3035, 3435, 3686, 4177, 4505, 5062

Offset: 0

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A308952, A308953, A308954, A308956, A308957, A308958, A308959, A308960.

Programs

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[m * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n,{7}],AllTrue[#,SquareFreeQ]&][[;;,6]]],{n,0,60}] (* Harvey P. Dale, Mar 07 2025 *)

Formula

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/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)^2 * m, where mu is the Möbius function (A008683).

a(n) = A308953(n) - A308954(n) - A308956(n) - A308957(n) - A308958(n) - A308959(n) - A308960(n).