A326531 Sum of the second largest parts of the partitions of n into 9 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 18, 22, 36, 45, 70, 86, 124, 148, 207, 252, 334, 396, 520, 609, 781, 907, 1144, 1321, 1653, 1906, 2344, 2687, 3278, 3746, 4533, 5143, 6175, 6983, 8305, 9337, 11037, 12362, 14493, 16168, 18831, 20956, 24264, 26876

Offset

0,12

Links

Table of n, a(n) for n=0..52.

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

a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326532(n).

Mathematica

Join[{0}, Table[Total[Select[IntegerPartitions[n, {9}], AllTrue[#, SquareFreeQ] &][[All, 2]]], {n, 60}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 08 2020 *)

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326525, A326526, A326527, A326528, A326529, A326530, A326532.

Sequence in context: A308959 A326451 A263982 * A326636 A210249 A233771

Adjacent sequences: A326528 A326529 A326530 * A326532 A326533 A326534

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 11 2019

Status

approved