cp's OEIS Frontend

A113414 Expansion of Sum_{k>0} x^k/(1-(-x^2)^k).

%I A113414 #9 Apr 30 2014 01:37:55
%S A113414 1,1,0,1,2,2,0,1,1,2,0,2,2,2,0,1,2,3,0,2,0,2,0,2,3,2,0,2,2,4,0,1,0,2,
%T A113414 0,3,2,2,0,2,2,4,0,2,2,2,0,2,1,3,0,2,2,4,0,2,0,2,0,4,2,2,0,1,4,4,0,2,
%U A113414 0,4,0,3,2,2,0,2,0,4,0,2,1,2,0,4,4,2,0,2,2,6,0,2,0,2,0,2,2,3,0,3,2,4,0,2,0
%N A113414 Expansion of Sum_{k>0} x^k/(1-(-x^2)^k).
%F A113414 Moebius transform is period 8 sequence [1, 0, -1, 0, 1, 2, -1, 0, ...].
%F A113414 G.f.: Sum_{k>0} x^k/(1-(-x^2)^k) = Sum_{k>0} x^k/(1+x^(2k))+2x^(6k)/(1-x^(8k)) = Sum_{k>0} -(-1)^k x^(2k-1)/(1+(-1)^k*x^(2k-1)).
%F A113414 a(4n+3) = 0.
%F A113414 a(n) = A001826(n) + (-1)^n * A001842(n). - _David Spies_, Sep 26 2012
%o A113414 (PARI) a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-4, d)+2*(n%2==0)*(d%4==3)))
%o A113414 (PARI) {a(n)=if(n<1, 0, if(n%4==3, 0, if(n%4==2, numdiv(n/2), if(n%4==0, sumdiv(n,d,d%2), sumdiv(n,d,(-1)^(d\2))))))}
%o A113414 (PARI) {a(n)=if(n<1, 0, polcoeff( sum(k=1,sqrtint(8*n+1)\2, (-1)^(k%4==2)*x^((k^2+k)/2)/(1-(-1)^(k\2)*x^k), x*O(x^n)), n))}
%o A113414 (PARI) {a(n)=if(n<1, 0, polcoeff( sum(k=1,n, x^k/(1-(-x^2)^k), x*O(x^n)), n))}
%Y A113414 A001227(n) = a(2*n), A008441(n) = a(4*n+1), A099774(n) = a(4*n+2).
%K A113414 nonn
%O A113414 1,5
%A A113414 _Michael Somos_, Oct 29 2005