A280127 Expansion of Product_{k>=2} 1/(1 - mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 7, 8, 11, 13, 16, 20, 24, 30, 35, 43, 52, 62, 74, 88, 104, 123, 146, 171, 201, 235, 275, 320, 373, 433, 502, 581, 672, 773, 891, 1024, 1176, 1348, 1543, 1764, 2013, 2296, 2614, 2974, 3378, 3833, 4345, 4920, 5565, 6288, 7098, 8005

Offset

0,6

Comments

Number of partitions of n into squarefree parts > 1 (A144338).

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.3 "Partitions into square-free parts", pp.351-352

Eric Weisstein's World of Mathematics, Squarefree

Index entries for related partition-counting sequences

Formula

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

Example

G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + 7*x^10 + ...
a(9) = 5 because we have [7, 2], [6, 3], [5, 2, 2], [3, 3, 3] and [3, 2, 2, 2].

Maple

with(numtheory): seq(coeff(series(mul(1/(1-mobius(k)^2*x^k), k=2..n), x, n+1), x, n), n=0..60); # Muniru A Asiru, Jul 30 2018

Mathematica

nmax = 65; CoefficientList[Series[Product[1/(1 - MoebiusMu[k]^2 x^k), {k, 2, nmax}], {x, 0, nmax}], x]

Prog

(PARI) {a(n) = if(n < 0, 0, polcoeff( 1 / prod(k=2, n, 1 - issquarefree(k)*x^k + x*O(x^n)), n))}; /* Michael Somos, Dec 26 2016 */

Crossrefs

Cf. A005117, A008683, A073576, A144338, A280128.

Sequence in context: A141472 A029034 A343941 * A237977 A115339 A305631

Adjacent sequences: A280124 A280125 A280126 * A280128 A280129 A280130

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 26 2016

Status

approved