A326629 Sum of the ninth largest parts of the partitions of n into 10 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 17, 25, 28, 38, 46, 61, 71, 90, 105, 137, 156, 194, 223, 278, 314, 381, 435, 526, 594, 706, 798, 951, 1064, 1246, 1398, 1641, 1822, 2117, 2360, 2735, 3030, 3480, 3856, 4427, 4884, 5555, 6140, 6984

Offset

0,13

Links

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

Index entries for sequences related to partitions

Formula

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} mu(r)^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-r)^2 * q, where mu is the Möbius function (A008683).

a(n) = A326627(n) - A326628(n) - A326630(n) - A326631(n) - A326632(n) - A326633(n) - A326634(n) - A326635(n) - A326636(n) - A326637(n).

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[q * MoebiusMu[r]^2 * 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 - r]^2 , {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

Crossrefs

Cf. A008683, A326626, A326627, A326628, A326630, A326631, A326632, A326633, A326634, A326635, A326636, A326637.

Sequence in context: A341153 A326628 A326524 * A326445 A069906 A304332

Adjacent sequences: A326626 A326627 A326628 * A326630 A326631 A326632

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 14 2019

Status

approved