OFFSET
0,2
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=x, y=2, z=A(x).
FORMULA
G.f. satisfies: A(x) = 1 + Sum_{n>=1} 2^n*A(x)*x^(2*n-1)/(1 - A(x)*x^(2*n-1)).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 6*x^2 + 26*x^3 + 106*x^4 + 474*x^5 + 2210*x^6 +...
which satisfies the following relations:
A(x) = 1 + A(x)*2*x/(1-2*x^2) + A(x)^2*2*x^2/(1-2*x^4) + A(x)^3*2*x^3/(1-2*x^6) +...
A(x) = 1 + 2*A(x)*x/(1-A(x)*x) + 4*A(x)*x^3/(1-A(x)*x^3) + 8*A(x)*x^5/(1-A(x)*x^5) +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, A^m*2*x^m/(1-2*x^(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, 2^m*A*x^(2*m-1)/(1-A*x^(2*m-1)+x*O(x^n)))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 30 2011
STATUS
approved