A308954 Sum of the smallest parts in the partitions of n into 7 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 10, 13, 15, 22, 25, 33, 37, 49, 55, 71, 77, 98, 109, 136, 148, 182, 199, 243, 264, 314, 344, 413, 441, 522, 567, 663, 711, 829, 896, 1036, 1106, 1269, 1370, 1572, 1666, 1903, 2041, 2316, 2460, 2780, 2971, 3350, 3546

Offset: 0

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A308952, A308953, A308955, A308956, A308957, A308958, A308959, A308960.

Programs

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[o * 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}]

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 * o, where mu is the Möbius function (A008683).

a(n) = A308953(n) - A308955(n) - A308956(n) - A308957(n) - A308958(n) - A308959(n) - A308960(n).