OFFSET
0,3
COMMENTS
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..140
FORMULA
G.f. satisfies: A(x) = Sum_{n>=1} ((1+x*A(x))^n - 1)^(n-1) / (1+x*A(x))^(n^2).
a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 2.341658334181687683758... . - Vaclav Kotesovec, Nov 08 2014
In closed form, c = 1 / (log(2) * sqrt(1-log(2)) * 2^((1-log(2))/2)). - Vaclav Kotesovec, May 03 2015
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 219*x^4 + 2227*x^5 + 26438*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x*A(x)/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)^2/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^3/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^4/(1+x*A(x))^16 +...
(2) A(x) = 1/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^2/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^3/(1+x*A(x))^16 +...
PROG
(PARI) {a(n)=local(q, A=1); for(i=1, n, q=1+x*A+x*O(x^n); A=sum(k=0, n+1, q^(-k^2)*(q^k-1)^k)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(q, A=1); for(i=1, n, q=1+x*A+x*O(x^n); A=sum(k=1, n+1, q^(-k^2)*(q^k-1)^(k-1))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 27 2012
STATUS
approved