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%I #21 Mar 12 2021 22:24:44
%S 16,128,704,3072,11488,38400,117632,335872,904784,2320128,5702208,
%T 13504512,30952544,68901888,149403264,316342272,655445792,1331327616,
%U 2655115712,5206288384,10049485312,19115905536,35867019904,66437873664
%N Expansion of -lambda(t + 1) in powers of the nome q = exp(Pi i t).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Seiichi Manyama, <a href="/A132136/b132136.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EllipticLambdaFunction.html">Elliptic Lambda Function</a>
%F Expansion of lambda(t) / ( 1 - lambda(t)) in powers of the nome q = exp(Pi i t).
%F 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.
%F Expansion of 16 * (eta(q^4) / eta(q))^8 in powers of q.
%F 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.
%F G.f.: 16 * x * (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^8.
%F a(n) = 16 * A092877(n) = -(-1)^n * A115977(n). a(n) = A014972(n) unless n=0.
%F Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = -1/2 + (3/8)*sqrt(2). - _Simon Plouffe_, Mar 04 2021
%e G.f. = 16*q + 128*q^2 + 704*q^3 + 3072*q^4 + 11488*q^5 + 38400*q^6 + 117632*q^7 + ...
%t a[ n_] := SeriesCoefficient[ With[ {m = InverseEllipticNomeQ@q}, m / (1 - m)], {q, 0, n}]; (* _Michael Somos_, Jun 03 2015 *)
%t a[ n_] := SeriesCoefficient[ 16 q (QPochhammer[ q^4] / QPochhammer[ q])^8, {q, 0, n}]; (* _Michael Somos_, Jun 03 2015 *)
%o (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))};
%o (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))};
%Y Cf. A014972, A092877, A115977.
%K nonn
%O 1,1
%A _Michael Somos_, Aug 11 2007
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