A326525 Sum of the eighth largest parts in the partitions of n into 9 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 18, 24, 28, 39, 46, 60, 69, 90, 105, 133, 149, 189, 216, 264, 297, 364, 412, 494, 553, 661, 743, 877, 972, 1149, 1280, 1493, 1650, 1922, 2126, 2454, 2702, 3107, 3429, 3916, 4291, 4895, 5374, 6086, 6647

Offset

0,12

Links

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

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

a(n) = A326523(n) - A326524(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).

Mathematica

Table[Total[Select[IntegerPartitions[n, {9}], AllTrue[#, SquareFreeQ]&][[;; , 8]]], {n, 0, 60}] (* Harvey P. Dale, Jan 30 2024 *)

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326526, A326527, A326528, A326529, A326530, A326531, A326532.

Sequence in context: A304332 A183564 A222707 * A326630 A317810 A308954

Adjacent sequences: A326522 A326523 A326524 * A326526 A326527 A326528

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 11 2019

Status

approved