A309480 Sum of the squarefree parts of the partitions of n into 5 parts.

0, 0, 0, 0, 0, 5, 6, 14, 20, 41, 62, 94, 124, 185, 245, 344, 425, 582, 736, 953, 1169, 1497, 1805, 2256, 2670, 3292, 3876, 4671, 5420, 6467, 7486, 8772, 10040, 11697, 13327, 15356, 17344, 19864, 22357, 25386, 28377, 32102, 35784, 40180, 44549, 49865, 55139

Offset

0,6

Links

David A. Corneth, Table of n, a(n) for n = 0..9999

Index entries for sequences related to partitions

Formula

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

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) |     62          94         124         185         245        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 12 2019

Mathematica

Table[Sum[Sum[Sum[Sum[i * MoebiusMu[i]^2 + j * MoebiusMu[j]^2 + k * MoebiusMu[k]^2 + l * MoebiusMu[l]^2 + (n - i - j - k - l) * 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}]

Crossrefs

Cf. A008683, A309457, A309478, A309479, A309480, A309481, A309482, A309484, A309485, A309486.

Sequence in context: A308839 A099330 A322479 * A180686 A308822 A289190

Adjacent sequences: A309477 A309478 A309479 * A309481 A309482 A309483

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 04 2019

Status

approved