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A280194
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Expansion of 1/(1 - Sum_{k>=1} mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).
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18
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1, 1, 2, 4, 7, 14, 27, 52, 100, 192, 370, 712, 1370, 2638, 5077, 9772, 18809, 36203, 69682, 134122, 258154, 496887, 956393, 1840836, 3543185, 6819813, 13126568, 25265616, 48630484, 93602468, 180163165, 346772545, 667457180, 1284701149, 2472753448, 4759480146, 9160901700, 17632623181, 33938733369, 65324235138, 125734088242
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OFFSET
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0,3
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COMMENTS
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Number of compositions (ordered partitions) into squarefree parts (A005117).
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LINKS
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FORMULA
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G.f.: 1/(1 - Sum_{k>=1} mu(k)^2*x^k).
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EXAMPLE
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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].
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
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1, add(
`if`(numtheory[issqrfree](j), a(n-j), 0), j=1..n))
end:
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MATHEMATICA
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nmax = 40; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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