A308843 Sum of the third largest parts in the partitions of n into 5 squarefree parts.

0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 12, 13, 21, 23, 32, 33, 49, 56, 77, 86, 117, 130, 162, 174, 223, 239, 295, 312, 391, 418, 497, 520, 631, 675, 801, 844, 1009, 1072, 1247, 1306, 1537, 1628, 1890, 1972, 2312, 2425, 2786, 2889, 3325, 3472, 3955, 4089, 4671, 4851, 5474

Offset

0,8

Comments

Conjecture: a(4*k + 3) < a(4*k + 4) for 4*k + 3 >= 195. This conjecture holds for all terms in the b-file. - David A. Corneth, Sep 16 2019

Links

David A. Corneth, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l)^2 * j, where mu is the Möbius function (A008683).

a(n) = A308839(n) - A308841(n) - A308842(n) - A308844(n) - A308845(n).

Mathematica

Table[Sum[Sum[Sum[Sum[j * MoebiusMu[l]^2*MoebiusMu[k]^2*MoebiusMu[j]^2 *MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l]^2, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n, {5}], AllTrue[#, SquareFreeQ]&][[All, 3]]], {n, 0, 60}] (* Harvey P. Dale, Dec 26 2022 *)

Crossrefs

Cf. A008683, A308839, A308840, A308841, A308842, A308844, A308845.

Sequence in context: A280094 A025053 A140496 * A086552 A098393 A263883

Adjacent sequences: A308840 A308841 A308842 * A308844 A308845 A308846

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 28 2019

Extensions

a(54)..a(55) from David A. Corneth, Sep 16 2019

Status

approved