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A280194 Expansion of 1/(1 - Sum_{k>=1} mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

%I A280194 #17 Feb 16 2025 08:33:38
%S A280194 1,1,2,4,7,14,27,52,100,192,370,712,1370,2638,5077,9772,18809,36203,
%T A280194 69682,134122,258154,496887,956393,1840836,3543185,6819813,13126568,
%U A280194 25265616,48630484,93602468,180163165,346772545,667457180,1284701149,2472753448,4759480146,9160901700,17632623181,33938733369,65324235138,125734088242
%N A280194 Expansion of 1/(1 - Sum_{k>=1} mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).
%C A280194 Number of compositions (ordered partitions) into squarefree parts (A005117).
%C A280194 INVERT transform of the absolute value of the Möbius function (A008966). - _Alois P. Heinz_, Feb 11 2021
%H A280194 Alois P. Heinz, <a href="/A280194/b280194.txt">Table of n, a(n) for n = 0..2000</a>
%H A280194 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H A280194 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A280194 G.f.: 1/(1 - Sum_{k>=1} mu(k)^2*x^k).
%e A280194 a(4) = 7 because we have [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].
%e A280194 G.f. = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 27*x^6 + 52*x^7 + ... - _Michael Somos_, Jul 13 2023
%p A280194 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A280194      `if`(numtheory[issqrfree](j), a(n-j), 0), j=1..n))
%p A280194     end:
%p A280194 seq(a(n), n=0..40);  # _Alois P. Heinz_, Feb 04 2021
%t A280194 nmax = 40; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%o A280194 (PARI) {a(n) = if(n<0, 0, polcoeff( 1/(1 - sum(k=1, n, x^k*abs(moebius(k)), x*O(x^n))), n, x))}; /* _Michael Somos_, Jul 13 2023 */
%Y A280194 Cf. A005117, A008683, A008966, A073576.
%K A280194 nonn
%O A280194 0,3
%A A280194 _Ilya Gutkovskiy_, Dec 28 2016