A326530 Sum of the third largest parts of the partitions of n into 9 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 14, 17, 27, 34, 48, 57, 84, 100, 138, 166, 225, 269, 354, 416, 540, 633, 796, 920, 1153, 1324, 1616, 1845, 2238, 2546, 3056, 3445, 4109, 4640, 5471, 6139, 7231, 8100, 9453, 10560, 12291, 13710, 15870, 17622, 20327

Offset

0,12

Links

Table of n, a(n) for n=0..53.

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

a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326531(n) - A326532(n).

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[j * 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, A326522, A326523, A326524, A326525, A326526, A326527, A326528, A326529, A326531, A326532.

Sequence in context: A378790 A308958 A326450 * A326635 A212214 A089426

Adjacent sequences: A326527 A326528 A326529 * A326531 A326532 A326533

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 11 2019

Status

approved