A280716 - OEIS

%I #6 Feb 16 2025 08:33:39

%S 1,0,0,1,0,1,0,1,1,0,1,1,1,1,1,2,2,1,3,2,3,3,3,4,4,3,5,4,5,6,4,8,6,8,

%T 8,9,11,10,11,14,13,14,15,16,19,16,20,22,22,23,26,29,30,31,35,39,38,

%U 43,44,49,50,52,58,59,64,67,71,77,82,85,93,97,107,108,117,125,131,138,143,157,162,168,179,194,199

%N Expansion of Product_{k>=2} (1 + mu(2*k-1)^2*x^(2*k-1)), where mu() is the Moebius function (A008683).

%C Number of partitions of n into distinct odd squarefree parts > 1.

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 16.4.3 "Partitions into square-free parts", pp.351-352

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F G.f.: Product_{k>=2} (1 + mu(2*k-1)^2*x^(2*k-1)).

%e a(18) = 3 because we have [15, 3], [13, 5] and [11, 7].

%t nmax = 84; CoefficientList[Series[Product[1 + MoebiusMu[2 k - 1]^2 x^(2 k - 1), {k, 2, nmax}], {x, 0, nmax}], x]

%Y Cf. A005117, A008683, A056911, A073576, A134337, A144338, A280128, A280169.

%K nonn

%O 0,16

%A _Ilya Gutkovskiy_, Jan 07 2017