A216878 G.f. satisfies: A(x) = 1 / Product_{n>=1} (1 + x^n*A(x)) * (1 + x^n/A(x)) * (1-x^n).

1, -1, 1, -3, 6, -17, 43, -125, 348, -1029, 3020, -9116, 27567, -84620, 260949, -812053, 2539208, -7989121, 25244540, -80136851, 255325972, -816447638, 2618870068, -8425244209, 27176810469, -87879769383, 284813417885, -925013053556, 3010106492409, -9813119711706

Offset: 0

Author

Paul D. Hanna, Sep 18 2012

Keywords

• sign

Examples

			G.f.: A(x) = 1 - x + x^2 - 3*x^3 + 6*x^4 - 17*x^5 + 43*x^6 - 125*x^7 +...
such that
1/A(x) = (1+x*A(x))*(1-x/A(x))*(1-x) * (1+x^2*A(x))*(1-x^2/A(x))*(1-x^2) * (1+x^3*A(x))*(1-x^3/A(x))*(1-x^3) * (1+x^4*A(x))*(1-x^4/A(x))*(1-x^4) *...
1/A(x) = (A(x) + x/A(x)) + (A(x)^2 + x^2/A(x)^2)*x + (A(x)^3 + x^3/A(x)^3)*x^3 + (A(x)^4 + x^4/A(x)^4)*x^6 + (A(x)^5 + x^5/A(x)^5)*x^10 +...
		

Crossrefs

Cf. A190791.

Programs

PARI

{a(n)=local(A=1-x);for(i=1,n,A=1/prod(m=1,n,(1+x^m/A)*(1+x^m*A)*(1-x^m)+x*O(x^n)));polcoeff(A,n)}

PARI

{a(n)=local(A=1-x);for(i=1,n,A=1/2*(A+1/sum(m=1,sqrtint(8*n+1),(A^m+x^m/A^m)*x^(m*(m-1)/2)+x*O(x^n))));polcoeff(A,n)}
for(n=0,21,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = 1 / Sum_{n>=1} (A(x)^n + x^n/A(x)^n) * x^(n*(n-1)/2) due to the Jacobi triple product identity.