A308844 Sum of the second largest parts in the partitions of n into 5 squarefree parts.

0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 16, 18, 29, 33, 52, 59, 83, 93, 125, 138, 178, 196, 252, 275, 350, 380, 471, 506, 634, 689, 839, 901, 1096, 1176, 1405, 1484, 1767, 1861, 2199, 2294, 2695, 2823, 3281, 3388, 3941, 4101, 4714, 4901, 5607, 5843, 6643, 6893

Offset

0,8

Links

Table of n, a(n) for n=0..52.

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

a(n) = A308839(n) - A308841(n) - A308842(n) - A308843(n) - A308845(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
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  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     10          16          18          29          33        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 16 2019

Mathematica

Table[Total[Select[IntegerPartitions[n, {5}], AllTrue[#, SquareFreeQ]&][[All, 2]]], {n, 0, 60}] (* Harvey P. Dale, Nov 19 2022 *)

Crossrefs

Cf. A008683, A308839, A308840, A308841, A308842, A308843, A308845.

Sequence in context: A226088 A026494 A043306 * A131355 A092534 A005232

Adjacent sequences: A308841 A308842 A308843 * A308845 A308846 A308847

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 28 2019

Status

approved