OFFSET
1,1
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function
FORMULA
Expansion of lambda(t) / ( 1 - lambda(t)) in powers of the nome q = exp(Pi i t).
Expansion of 16 * q * (psi(q^2) / phi(-q))^4 = 16 * q * (psi(q^2) / psi(-q))^8 = 16 * q * (psi(q) / phi(-q^2))^8 = 16 * q * (psi(-q) / phi(-q))^8 = 16 * q * (f(-q^4) / f(-q))^8 = 16 * q / (chi(-q) * chi(-q^2))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of 16 * (eta(q^4) / eta(q))^8 in powers of q.
Given G.f. A(x), then B(x) = A(x) / 16 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v - 16*u*v - 16*v^2 - 256*u*v^2.
G.f.: 16 * x * (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^8.
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = -1/2 + (3/8)*sqrt(2). - Simon Plouffe, Mar 04 2021
EXAMPLE
G.f. = 16*q + 128*q^2 + 704*q^3 + 3072*q^4 + 11488*q^5 + 38400*q^6 + 117632*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ With[ {m = InverseEllipticNomeQ@q}, m / (1 - m)], {q, 0, n}]; (* Michael Somos, Jun 03 2015 *)
a[ n_] := SeriesCoefficient[ 16 q (QPochhammer[ q^4] / QPochhammer[ q])^8, {q, 0, n}]; (* Michael Somos, Jun 03 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); 16 * polcoeff( (eta(x^4 + A) / eta(x + A))^8, n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^8 - 1, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 11 2007
STATUS
approved