A192399 G.f. A(x) satisfies: A(x) = 1 + Sum_{n>=1} x^n * A(x)^n/(1 - x*A(x)^(2*n)).

1, 1, 3, 11, 48, 233, 1218, 6722, 38668, 229864, 1403618, 8766186, 55818141, 361499355, 2376956264, 15845876429, 106988044753, 731026642533, 5051920683481, 35296182297157, 249249589433312, 1778775804736254, 12828718640894604

Offset: 0

Author

Paul D. Hanna, Jun 30 2011

Keywords

• nonn

Comments

Related q-series identity:

Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)); here q=A(x), y=x, z=x.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 48*x^4 + 233*x^5 + 1218*x^6 +...
which satisfies the following relations:
A(x) = 1 + x*A(x)/(1-x*A(x)^2) + x^2*A(x)^2/(1-x*A(x)^4) + x^3*A(x)^3/(1-x*A(x)^6) +...
A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)^3/(1-x*A(x)^3) + x^3*A(x)^5/(1-x*A(x)^5) +...
A(x) = 1 + x*A(x) + x^2*A(x)^3*(1 + 1/A(x)) + x^3*A(x)^6*(1 + 1/A(x) + 1/A(x)^3) + x^4*A(x)^10*(1 + 1/A(x) + 1/A(x)^3 + 1/A(x)^6) +...
		

Crossrefs

Cf. A192400, A192401, A192405.

Programs

PARI

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

PARI

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

PARI

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

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n-1)/(1 - x*A(x)^(2*n-1)).

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^(n*(n+1)/2) * Sum_{k=0..n-1} A(x)^(-k*(k+1)/2). - Paul D. Hanna, Jul 01 2011