A308845 Sum of the largest parts in the partitions of n into 5 squarefree parts.

0, 0, 0, 0, 0, 1, 2, 5, 5, 13, 19, 32, 35, 52, 66, 96, 104, 147, 177, 240, 263, 356, 392, 514, 543, 712, 763, 972, 1015, 1271, 1370, 1652, 1728, 2084, 2223, 2623, 2750, 3275, 3480, 4087, 4276, 5011, 5312, 6141, 6388, 7443, 7842, 8987, 9405, 10753, 11335

Offset

0,7

Links

Table of n, a(n) for n=0..50.

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 * (n-i-j-k-l), where mu is the Möbius function (A008683).

a(n) = A308839(n) - A308841(n) - A308842(n) - A308843(n) - A308844(n).

Example

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) |     19          32          35          52          66        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 16 2019

Mathematica

Table[Sum[Sum[Sum[Sum[(n - i - j - k - l) * 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, 80}]

Crossrefs

Cf. A008683, A308839, A308840, A308841, A308842, A308843, A308844.

Sequence in context: A176168 A308770 A222114 * A308911 A308960 A326452

Adjacent sequences: A308842 A308843 A308844 * A308846 A308847 A308848

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 28 2019

Status

approved