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Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the larger part squarefree.
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%I #20 Jan 19 2021 15:54:10

%S 0,2,4,12,6,20,26,46,52,58,66,96,80,90,104,148,162,210,224,276,290,

%T 346,318,382,352,372,394,416,438,518,542,566,592,684,712,810,838,866,

%U 898,1008,960,1076,1112,1234,1270,1306,1344,1476,1514,1454,1494,1636,1676

%N Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the larger part squarefree.

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

%F a(n) = 2 * Sum_{i=1..n} (n - i) * mu(2*n - i)^2, where mu is the Möbius function (A008683).

%t Table[2*Sum[(n - i) MoebiusMu[2 n - i]^2, {i, n}], {n, 80}]

%t Table[Total[#[[1]]-#[[2]]&/@Select[IntegerPartitions[2n,{2}],SquareFreeQ[ #[[1]]]&]],{n,60}] (* _Harvey P. Dale_, Jan 19 2021 *)

%o (PARI) a(n) = 2*sum(i=1, n, (n-i)*moebius(2*n-i)^2); \\ _Michel Marcus_, Nov 08 2017

%Y Cf. A008683, A008966, A294062.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Oct 22 2017