A308957 - OEIS

%I #8 Sep 24 2023 12:07:52

%S 0,0,0,0,0,0,0,1,1,2,2,5,7,11,14,20,24,36,41,56,64,86,98,129,147,193,

%T 222,284,324,409,457,567,635,773,862,1037,1147,1375,1516,1778,1953,

%U 2290,2510,2920,3186,3680,4017,4614,4996,5734,6226,7081,7682,8732,9450

%N Sum of the fourth largest parts in the partitions of n into 7 squarefree parts.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F 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 * k, where mu is the Möbius function (A008683).

%F a(n) = A308953(n) - A308954(n) - A308955(n) - A308956(n) - A308958(n) - A308959(n) - A308960(n).

%t Table[Sum[Sum[Sum[Sum[Sum[Sum[k * 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}]

%t Table[Total[Select[IntegerPartitions[n,{7}],AllTrue[#,SquareFreeQ]&][[;;,4]]],{n,0,60}] (* _Harvey P. Dale_, Sep 24 2023 *)

%Y Cf. A008683, A308952, A308953, A308954, A308955, A308956, A308958, A308959, A308960.

%K nonn

%O 0,10

%A _Wesley Ivan Hurt_, Jul 03 2019