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

1,11

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000931, A113788.

Sequence in context: A036816 A367399 A113788 * A127207 A173513 A367693

Adjacent sequences: A018240 A018241 A018242 * A018244 A018245 A018246

Keyword

nonn,nice

Author

N. J. A. Sloane, David Broadhurst

Extensions

More terms from Joerg Arndt, Jul 18 2012

Status

approved