A326626 Number of 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, 16, 23, 26, 35, 41, 54, 62, 79, 90, 115, 130, 161, 182, 224, 251, 303, 341, 408, 456, 539, 601, 709, 786, 915, 1014, 1179, 1299, 1496, 1649, 1892, 2078, 2368, 2597, 2953, 3230, 3645, 3986, 4492, 4895

Offset: 0

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

• nonn

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999
• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A326627, A326628, A326629, A326630, A326631, A326632, A326633, A326634, A326635, A326636, A326637.

Programs

Mathematica

Table[Count[IntegerPartitions[n,{10}],?(AllTrue[#,SquareFreeQ]&)],{n,0,60}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Aug 25 2019 *)

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, where mu is the Möbius function (A008683).

a(n) = A326627(n)/n for n > 0.