A294063 Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the larger part squarefree.
0, 2, 4, 12, 6, 20, 26, 46, 52, 58, 66, 96, 80, 90, 104, 148, 162, 210, 224, 276, 290, 346, 318, 382, 352, 372, 394, 416, 438, 518, 542, 566, 592, 684, 712, 810, 838, 866, 898, 1008, 960, 1076, 1112, 1234, 1270, 1306, 1344, 1476, 1514, 1454, 1494, 1636, 1676
Offset: 1
Programs
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Mathematica
Table[2*Sum[(n - i) MoebiusMu[2 n - i]^2, {i, n}], {n, 80}] Table[Total[#[[1]]-#[[2]]&/@Select[IntegerPartitions[2n,{2}],SquareFreeQ[ #[[1]]]&]],{n,60}] (* Harvey P. Dale, Jan 19 2021 *) -
PARI
a(n) = 2*sum(i=1, n, (n-i)*moebius(2*n-i)^2); \\ Michel Marcus, Nov 08 2017
Formula
a(n) = 2 * Sum_{i=1..n} (n - i) * mu(2*n - i)^2, where mu is the Möbius function (A008683).