OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..250
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} (x*A(x))^n / (1 - x^(2*n+1)/A(x)).
(2) A(x) = Sum_{n>=0} (x/A(x))^n / (1 - x^(2*n+1)*A(x)).
(3) A(x) = Sum_{n>=0} x^(2*n*(n+1)) * (1 - x^(4*n+2)) / ((1 - x^(2*n+1)*A(x))*(1 - x^(2*n+1)/A(x)).
a(n) ~ c * d^n / n^(3/2), where d = 4.24149290765489... and c = 0.598891272666... - Vaclav Kotesovec, Mar 03 2021
EXAMPLE
G.f.: A(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 19*x^4 + 58*x^5 + 198*x^6 + 680*x^7 + 2410*x^8 + 8695*x^9 + 31870*x^10 + 118377*x^11 + 444315*x^12 + ...
where
A(x) = 1/(1 - x/A(x)) + x*A(x)/(1 - x^3/A(x)) + x^2*A(x)^2/(1 - x^5/A(x)) + x^3*A(x)^3/(1 - x^7/A(x)) + x^4*A(x)^4/(1 - x^9/A(x)) + ...
also
A(x) = 1/(1 - x*A(x)) + (x/A(x))/(1 - x^3*A(x)) + (x/A(x))^2/(1 - x^5*A(x)) + (x/A(x))^3/(1 - x^7*A(x)) + (x/A(x))^4/(1 - x^9*A(x)) + ...
PROG
(PARI) {a(n) = my(A=1); for(i=1, n,
A = sum(m=0, n, x^m*A^m/(1 - x^(2*m+1)/A +x*O(x^n)) ); );
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n,
A = sum(m=0, n, x^m/A^m/(1 - x^(2*m+1)*A +x*O(x^n)) ); );
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n,
A = sum(m=0, sqrtint(n), x^(2*m*(m+1)) * (1 - x^(4*m+2)) / ((1 - x^(2*m+1)*A)*(1 - x^(2*m+1)/A) +x*O(x^n)) ); );
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 25 2021
STATUS
approved