A326445 Sum of the smallest parts of the partitions of n into 8 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 16, 22, 26, 35, 40, 52, 59, 78, 86, 108, 122, 153, 169, 207, 231, 280, 310, 371, 409, 487, 535, 630, 688, 812, 883, 1028, 1119, 1295, 1409, 1619, 1754, 2014, 2180, 2479, 2679, 3046, 3284, 3707, 3994, 4502

Offset

0,11

Links

Table of n, a(n) for n=0..56.

Index entries for sequences related to partitions

Formula

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

a(n) = A326444(n) - A326446(n) - A326447(n) - A326448(n) - A326449(n) - A326450(n) - A326451(n) - A326452(n).

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[p * 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, A326444, A326446, A326447, A326448, A326449, A326450, A326451, A326452.

Sequence in context: A326628 A326524 A326629 * A069906 A304332 A183564

Adjacent sequences: A326442 A326443 A326444 * A326446 A326447 A326448

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 06 2019

Status

approved