OFFSET
0,3
COMMENTS
The g.f. A(x) of this sequence is motivated by the following identity:
Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1 - q*r^n)) ;
here, p = x, q = x^2, and r = A(x).
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n / (1 - x^2*A(x)^n).
(2) A(x) = Sum_{n>=0} x^(2*n) / (1 - x*A(x)^n).
(3) A(x) = Sum_{n>=0} x^(3*n) * A(x)^(n^2) * (1 - x^3*A(x)^(2*n)) / ((1 - x*A(x)^n)*(1 - x^2*A(x)^n)).
EXAMPLE
G.f.: A(x) = = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 16*x^6 + 38*x^7 + 97*x^8 + 257*x^9 + 704*x^10 + 1985*x^11 + 5742*x^12 + ...
where
A(x) = 1/(1 - x^2) + x/(1 - x^2*A(x)) + x^2/(1 - x^2*A(x)^2) + x^3/(1 - x^2*A(x)^3) + x^4/(1 - x^2*A(x)^4) + ...
also
A(x) = 1/(1 - x) + x^2/(1 - x*A(x)) + x^4/(1 - x*A(x)^2) + x^6/(1 - x*A(x)^3) + x^8/(1 - x*A(x)^4) + ...
PROG
(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^m /(1 - x^2*A^m +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^(2*m) /(1 - x*A^m +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 11 2021
STATUS
approved