A326444 Sum of all the parts in the partitions of n into 8 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 8, 9, 20, 22, 48, 65, 112, 135, 208, 255, 378, 456, 640, 756, 1034, 1219, 1632, 1875, 2444, 2835, 3640, 4147, 5220, 5952, 7392, 8382, 10234, 11550, 14004, 15688, 18810, 21021, 25040, 27798, 32802, 36421, 42680, 47160, 54832, 60489

Offset

0,9

Links

Table of n, a(n) for n=0..47.

Index entries for sequences related to partitions

Formula

a(n) = n * Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2, where mu is the Möbius function (A008683).

a(n) = n * A326443(n).

a(n) = A326445(n) + A326446(n) + A326447(n) + A326448(n) + A326449(n) + A326450(n) + A326451(n) + A326452(n).

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[n * MoebiusMu[p]^2 * 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 - p]^2, {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]

Crossrefs

Cf. A008683, A326443, A326445, A326446, A326447, A326448, A326449, A326450, A326451, A326452.

Sequence in context: A240915 A281225 A071869 * A380825 A309484 A308989

Adjacent sequences: A326441 A326442 A326443 * A326445 A326446 A326447

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 06 2019

Status

approved