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

0,3

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

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

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

Cf. A005117, A008683, A008966, A073576.

Sequence in context: A123196 A347783 A079968 * A001631 A108758 A018085

Adjacent sequences: A280191 A280192 A280193 * A280195 A280196 A280197

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 28 2016

Status

approved