A018243 Inverse Euler transform of A000931.

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 11, 13, 17, 21, 28, 34, 45, 56, 73, 92, 120, 151, 197, 250, 324, 414, 537, 687, 892, 1145, 1484, 1911, 2479, 3196, 4148, 5359, 6954, 9000, 11687, 15140, 19672, 25516, 33166, 43065, 56010, 72784, 94716, 123185, 160380, 208740, 271913, 354123, 461529, 601436, 784209, 1022505, 1333856

Offset: 1

Author

N. J. A. Sloane, David Broadhurst

Keywords

• nonn
• nice

Examples

			x^3 + x^5 + x^7 + x^8 + x^9 + x^10 + 2*x^11 + 2*x^12 + 3*x^13 + 3*x^14 + ...
		

Links

• Joerg Arndt, Table of n, a(n) for n = 1..1000
• D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B 393, No.3-4, 403-412 (1997).
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000931, A113788.

Programs

Maple

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000931):
seq(a(n), n = 1..65); # Peter Luschny, Nov 21 2022

Mathematica

a[n_] := (1/n)*Sum[ MoebiusMu[n/d]*Floor[ Re[ N[ RootSum[ -1-#+#^3&, #^d& ]]]] , {d, Divisors[n]}]; a[2]=0; Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 05 2012, after Michael Somos *)

Sage

z = PowerSeriesRing(ZZ, 'z').gen().O(30)
r = (1 - (z**2 + z**3))/(1 - z**2)
F = -z*r.derivative()/r
[sum(moebius(n//d)*F[d] for d in divisors(n))//n for n in range(1, 24)] # F. Chapoton, Apr 25 2020

Formula

a(n) = A113788(n) unless n=2. - Michael Somos, Apr 06 2012

Reciprocal of g.f. of A000931 = (1 - x^2 - x^3) / (1 - x^2) = 1 - x^3 - x^5 - x^7 - x^9 - ... = Product_{k>0} (1 - x^k)^a(n). - Michael Somos, Jul 17 2012

a(n) ~ A060006^n / n. - Vaclav Kotesovec, Oct 09 2019

Extensions

More terms from Joerg Arndt, Jul 18 2012