OFFSET
0,3
COMMENTS
Compare g.f. to a g.f. C(x) of the Catalan sequence:
C(x) = Sum_{n>=0} x^n*(1 + x*C(x)^2)^n where C(x) = 1 + x*C(x)^2.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. satisfies: A(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2) / (1 - x*A(x)^n)^(n+1). - Paul D. Hanna, Sep 24 2014
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 35*x^5 + 123*x^6 +...
such that
A(x) = 1 + x*(1+x*A(x)) + x^2*(1+x*A(x)^2)^2 + x^3*(1+x*A(x)^3)^3 + x^4*(1+x*A(x)^4)^4 + x^5*(1+x*A(x)^5)^5 + x^6*(1+x*A(x)^6)^6 +...
The g.f. satisfies the series identity:
A(x) = 1/(1-x) + x^2*A(x)/(1-x*A(x))^2 + x^4*A(x)^4/(1-x*A(x)^2)^3 + x^6*A(x)^9/(1-x*A(x)^3)^4 + x^8*A(x)^16/(1-x*A(x)^4)^5 + x^10*A(x)^25/(1-x*A(x)^5)^6 +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*(1+x*(A+x*O(x^n))^m)^m)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, x^(2*k)*A^(k^2)/(1 - x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 24 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 01 2011
STATUS
approved