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URL: https://oeis.org/A280194

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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).
18
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
OFFSET
0,3
COMMENTS
Number of compositions (ordered partitions) into squarefree parts (A005117).
INVERT transform of the absolute value of the Möbius function (A008966). - Alois P. Heinz, Feb 11 2021
LINKS
FORMULA
G.f.: 1/(1 - Sum_{k>=1} mu(k)^2*x^k).
EXAMPLE
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
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(
`if`(numtheory[issqrfree](j), a(n-j), 0), j=1..n))
end:
seq(a(n), n=0..40); # Alois P. Heinz, Feb 04 2021
MATHEMATICA
nmax = 40; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 28 2016
STATUS
approved