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

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, 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, 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

Offset: 1

Author

Michael Somos, Oct 29 2005

Keywords

• nonn

Crossrefs

A001227(n) = a(2*n), A008441(n) = a(4*n+1), A099774(n) = a(4*n+2).

Programs

PARI

a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-4, d)+2*(n%2==0)*(d%4==3)))

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))))))}

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))}

PARI

{a(n)=if(n<1, 0, polcoeff( sum(k=1,n, x^k/(1-(-x^2)^k), x*O(x^n)), n))}

Formula

Moebius transform is period 8 sequence [1, 0, -1, 0, 1, 2, -1, 0, ...].

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)).

a(4n+3) = 0.

a(n) = A001826(n) + (-1)^n * A001842(n). - David Spies, Sep 26 2012