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

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

Author

Ilya Gutkovskiy, Dec 28 2016

Keywords

• nonn

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

Examples

			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
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000
• Eric Weisstein's World of Mathematics, Squarefree
• Index entries for sequences related to compositions

Crossrefs

Cf. A005117, A008683, A008966, A073576.

Programs

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]

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 */

Formula

G.f.: 1/(1 - Sum_{k>=1} mu(k)^2*x^k).