A132136 Expansion of -lambda(t + 1) in powers of the nome q = exp(Pi i t).

16, 128, 704, 3072, 11488, 38400, 117632, 335872, 904784, 2320128, 5702208, 13504512, 30952544, 68901888, 149403264, 316342272, 655445792, 1331327616, 2655115712, 5206288384, 10049485312, 19115905536, 35867019904, 66437873664

Offset

1,1

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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.

a(n) = 16 * A092877(n) = -(-1)^n * A115977(n). a(n) = A014972(n) unless n=0.

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

Cf. A014972, A092877, A115977.

Sequence in context: A014972 A115977 A128692 * A163399 A067488 A308310

Adjacent sequences: A132133 A132134 A132135 * A132137 A132138 A132139

Keyword

nonn

Author

Michael Somos, Aug 11 2007

Status

approved