A117209 - OEIS

%I #25 Jul 06 2024 14:14:54

%S 1,1,0,-1,-1,-1,0,0,0,0,1,0,0,-1,0,1,2,-1,-1,-2,0,1,3,-1,0,-1,1,-1,1,

%T -3,1,-1,1,-2,3,0,6,-1,-1,-6,2,-4,4,-3,2,-4,6,-5,6,-2,7,-5,4,-13,5,-3,

%U 11,-6,8,-14,10,-6,9,-14,11,-14,15,-13,9,-15,24,-13,19,-21,12,-20,27,-24,21,-26,22,-24,33,-33,32,-26

%N G.f. A(x) satisfies 1/(1-x) = Product_{k>=1} A(x^k).

%C Self-convolution inverse is A117208.

%H Vaclav Kotesovec, <a href="/A117209/b117209.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Paul D. Hanna)

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F G.f.: A(x) = exp( Sum_{n>=1} A023900(n)*x^n/n ), where A023900 is the Dirichlet inverse of Euler totient function.

%F Euler transform of the Möbius function A008683. - _Stuart Clary_, _Franklin T. Adams-Watters_ and _Vladeta Jovovic_, Apr 15 2006

%F G.f.: A(x) = Product_{k>=1}(1 - x^k)^(-mu(k)) where mu(k) is the Möbius function, A008683. - _Stuart Clary_ and _Franklin T. Adams-Watters_, Apr 15 2006

%F G.f.: A(x) = Product_{k>=1} (1 + x^(2*k-1))^mu(2*k-1), where mu() is the Moebius function. - _Seiichi Manyama_, Jul 06 2024

%t nmax = 85; CoefficientList[ Series[ Product[ (1 - x^k)^(-MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* _Stuart Clary_, Apr 15 2006 *)

%o (PARI) {a(n)=polcoeff(exp(sum(k=1,n+1,sumdiv(k,d,d*moebius(d))*x^k/k)+x*O(x^n)),n)}

%Y Cf. A023900 (l.g.f.), A117208 (inverse); variants: A117210, A117211, A117212.

%Y Cf. A008683.

%K sign

%O 0,17

%A _Paul D. Hanna_, Mar 03 2006