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Sum of the second largest parts in the partitions of n into 5 squarefree parts.
4

%I #14 Nov 19 2022 21:43:04

%S 0,0,0,0,0,1,1,3,4,8,10,16,18,29,33,52,59,83,93,125,138,178,196,252,

%T 275,350,380,471,506,634,689,839,901,1096,1176,1405,1484,1767,1861,

%U 2199,2294,2695,2823,3281,3388,3941,4101,4714,4901,5607,5843,6643,6893

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F 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).

%F a(n) = A308839(n) - A308841(n) - A308842(n) - A308843(n) - A308845(n).

%e The partitions of n into 5 parts for n = 10, 11, ..

%e 1+1+1+1+10

%e 1+1+1+2+9

%e 1+1+1+3+8

%e 1+1+1+4+7

%e 1+1+1+5+6

%e 1+1+1+1+9 1+1+2+2+8

%e 1+1+1+2+8 1+1+2+3+7

%e 1+1+1+3+7 1+1+2+4+6

%e 1+1+1+4+6 1+1+2+5+5

%e 1+1+1+5+5 1+1+3+3+6

%e 1+1+1+1+8 1+1+2+2+7 1+1+3+4+5

%e 1+1+1+2+7 1+1+2+3+6 1+1+4+4+4

%e 1+1+1+3+6 1+1+2+4+5 1+2+2+2+7

%e 1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6

%e 1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5

%e 1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e --------------------------------------------------------------------------

%e n | 10 11 12 13 14 ...

%e --------------------------------------------------------------------------

%e a(n) | 10 16 18 29 33 ...

%e --------------------------------------------------------------------------

%e - _Wesley Ivan Hurt_, Sep 16 2019

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

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

%K nonn

%O 0,8

%A _Wesley Ivan Hurt_, Jun 28 2019