A326522 Number of partitions of n into 9 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 16, 22, 25, 34, 39, 51, 58, 74, 84, 106, 118, 147, 165, 201, 223, 270, 301, 358, 396, 468, 518, 607, 666, 776, 853, 985, 1077, 1239, 1354, 1546, 1684, 1915, 2086, 2360, 2560, 2886, 3132, 3513, 3800, 4251

Offset

0,12

Links

Table of n, a(n) for n=0..57.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^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-q)^2, where mu is the Möbius function (A008683).

a(n) = A326523(n)/n for n > 0.

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[MoebiusMu[q]^2 * 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 - q]^2, {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 80}]

Crossrefs

Cf. A008683, A326523, A326524, A326525, A326526, A326527, A326528, A326529, A326530, A326531, A326532.

Sequence in context: A308952 A302401 A326443 * A326626 A341153 A326628

Adjacent sequences: A326519 A326520 A326521 * A326523 A326524 A326525

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 11 2019

Status

approved