OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..100
FORMULA
G.f.: A(x) = A = 1 + x/(2 - A/(1 - A*(A^2-1)/(2 - A^5/(1 - A^3*(A^4-1)/(2 - A^9/(1 - A^5*(A^6-1)/(2 - A^13/(1 - A^7*(A^8-1)/(2 - ...))))))))), a continued fraction due to a partial elliptic theta function identity.
G.f.: A(x) = 1 + x/2 * Sum_{n>=0} A(x)^n / 2^n * Product_{k=1..n} (2 - A(x)^(4*k-3)) / (2 - A(x)^(4*k-1)), due to a q-series identity.
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 45*x^3 + 1095*x^4 + 35151*x^5 + 1381725*x^6 + 64175631*x^7 + 3451040205*x^8 + 211905683649*x^9 + ...
such that
A(x) = 1 + x/2 * (A(x)/2 + A(x)^4/2^2 + A(x)^9/2^3 + A(x)^16/2^4 + A(x)^25/2^5 + A(x)^36/2^6 + A(x)^49/2^7 + ... + A(x)^(n^2)/2^n + ...).
PROG
(PARI) /* Continued fraction expression: */
{a(n) = my(CF=1, A); for(i=0, n, A = 1 + x*CF +x*O(x^n); for(k=0, n, CF = 1/(2 - A^(4*n-4*k+1)/(1 - A^(2*n-2*k+1)*(A^(2*n-2*k+2) - 1)*CF)) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 05 2018
STATUS
approved