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%I #5 Feb 06 2021 08:53:37
%S 1,1,2,2,4,5,8,9,13,16,23,27,37,44,59,70,91,107,138,162,205,240,300,
%T 350,432,504,615,715,865,1003,1206,1394,1663,1918,2276,2618,3088,3544,
%U 4160,4764,5566,6360,7401,8440,9779,11131,12854,14602,16803,19051,21859,24741,28303
%N Number of partitions of 2*n into exactly n squarefree parts.
%F G.f.: Product_{k>=1} 1 / (1 - mu(k+1)^2 * x^k).
%t nmax = 52; CoefficientList[Series[Product[1/(1 - MoebiusMu[k + 1]^2 x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d MoebiusMu[d + 1]^2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]
%Y Cf. A005117, A008683, A073576, A341071, A341154.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Feb 06 2021
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