A326443 Number of partitions of n into 8 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 15, 21, 24, 32, 36, 47, 53, 68, 75, 94, 105, 130, 143, 174, 192, 231, 254, 301, 330, 389, 424, 495, 539, 626, 678, 781, 847, 970, 1048, 1192, 1287, 1461, 1572, 1772, 1908, 2144, 2301, 2573, 2762, 3079, 3295

Offset: 0

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A326444, A326445, A326446, A326447, A326448, A326449, A326450, A326451, A326452.

Programs

Mathematica

Table[Count[IntegerPartitions[n,{8}],?(AllTrue[#,SquareFreeQ]&)],{n,0,60}] (* _Harvey P. Dale, Mar 10 2023 *)

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, where mu is the Möbius function (A008683).

a(n) = A326444(n)/n for n > 0.