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A280127 Expansion of Product_{k>=2} 1/(1 - mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

%I A280127 #13 Feb 16 2025 08:33:38
%S A280127 1,0,1,1,1,2,3,3,4,5,7,8,11,13,16,20,24,30,35,43,52,62,74,88,104,123,
%T A280127 146,171,201,235,275,320,373,433,502,581,672,773,891,1024,1176,1348,
%U A280127 1543,1764,2013,2296,2614,2974,3378,3833,4345,4920,5565,6288,7098,8005
%N A280127 Expansion of Product_{k>=2} 1/(1 - mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).
%C A280127 Number of partitions of n into squarefree parts > 1 (A144338).
%H A280127 G. C. Greubel, <a href="/A280127/b280127.txt">Table of n, a(n) for n = 0..5000</a>
%H A280127 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 A280127 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H A280127 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A280127 G.f.: Product_{k>=2} 1/(1 - mu(k)^2*x^k).
%e A280127 G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + 7*x^10 + ...
%e A280127 a(9) = 5 because we have [7, 2], [6, 3], [5, 2, 2], [3, 3, 3] and [3, 2, 2, 2].
%p A280127 with(numtheory): seq(coeff(series(mul(1/(1-mobius(k)^2*x^k),k=2..n), x,n+1),x,n),n=0..60); # _Muniru A Asiru_, Jul 30 2018
%t A280127 nmax = 65; CoefficientList[Series[Product[1/(1 - MoebiusMu[k]^2 x^k), {k, 2, nmax}], {x, 0, nmax}], x]
%o A280127 (PARI) {a(n) = if(n < 0, 0, polcoeff( 1 / prod(k=2, n, 1 - issquarefree(k)*x^k + x*O(x^n)), n))}; /* _Michael Somos_, Dec 26 2016 */
%Y A280127 Cf. A005117, A008683, A073576, A144338, A280128.
%K A280127 nonn
%O A280127 0,6
%A A280127 _Ilya Gutkovskiy_, Dec 26 2016