A308842 Sum of the fourth largest parts in the partitions of n into 5 squarefree parts.

0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 7, 9, 10, 16, 18, 25, 25, 35, 40, 51, 54, 74, 83, 105, 112, 145, 156, 191, 198, 246, 267, 317, 331, 402, 430, 502, 520, 613, 652, 758, 791, 925, 979, 1118, 1152, 1337, 1406, 1603, 1667, 1905, 2009, 2266, 2343, 2652, 2787, 3134

Offset: 0

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

• nonn

Examples

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |      7           9          10          16          18        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 16 2019
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A308839, A308840, A308841, A308843, A308844, A308845.

Programs

Mathematica

Table[Sum[Sum[Sum[Sum[k * 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}]

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

a(n) = A308839(n) - A308841(n) - A308843(n) - A308844(n) - A308845(n).