A308957 Sum of the fourth largest parts in the partitions of n into 7 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 7, 11, 14, 20, 24, 36, 41, 56, 64, 86, 98, 129, 147, 193, 222, 284, 324, 409, 457, 567, 635, 773, 862, 1037, 1147, 1375, 1516, 1778, 1953, 2290, 2510, 2920, 3186, 3680, 4017, 4614, 4996, 5734, 6226, 7081, 7682, 8732, 9450

Offset

0,10

Links

Table of n, a(n) for n=0..54.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3} Sum_{i=j..floor((n-j-k-l-m-o)/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)^2 * k, where mu is the Möbius function (A008683).

a(n) = A308953(n) - A308954(n) - A308955(n) - A308956(n) - A308958(n) - A308959(n) - A308960(n).

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[k * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n, {7}], AllTrue[#, SquareFreeQ]&][[;; , 4]]], {n, 0, 60}] (* Harvey P. Dale, Sep 24 2023 *)

Crossrefs

Cf. A008683, A308952, A308953, A308954, A308955, A308956, A308958, A308959, A308960.

Sequence in context: A308908 A259446 A265769 * A240488 A238491 A326449

Adjacent sequences: A308954 A308955 A308956 * A308958 A308959 A308960

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 03 2019

Status

approved