OFFSET
-1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This series is also called Weber's modular function. - N. J. A. Sloane, Jun 23 2011
Or, better, it is the 24th power of Weber's modular function f(). - Michael Somos, Jan 10 2017
Given g.f. A(q), Greenhill (1895) denotes 1/64 * A(q) by tau_1 on page 409 equation (43). - Michael Somos, Jul 17 2013
REFERENCES
S. Ramanujan, Modular Equations and Approximations to Pi, pp. 23-39 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 26.
LINKS
T. D. Noe, Table of n, a(n) for n = -1..1000
Barry Brent, Finite Field Models of Polynomials Interpolating Fourier Coefficients of Modular Functions for Hecke Groups, Integers (2024) Vol. 24, Art. No. A18.
A. G. Greenhill, The Transformation and Division of Elliptic Functions, Proceedings of the London Mathematical Society (1895) 403-486.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Titus Piezas III, Pi Formulas, Ramanujan, and the Baby Monster Group
Titus Piezas III, Ramanujan's Constant exp(Pi sqrt(163)) And Its Cousins
Titus Piezas III, 0011: Article 1 (The j-function) - A collection of Algebraic Identities
Titus Piezas III, 0022: The 163 Dimensions - A collection of Algebraic Identities
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function
FORMULA
Expansion of q^(-1) * chi(q)^24 where chi() is a Ramanujan theta function.
Expansion of (eta(q^2)^2 / (eta(q) * eta(q^4)))^24 in powers of q.
Euler transform of period 4 sequence [ 24, -24, 24, 0, ...].
G.f. is Fourier series of a level 4 modular function. f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (u^3 + v^3) + (-u^3 + 48*u^2 - 96*u) * v^3 + (48*u^3 + 1791*u^2 + 2352*u) * v^2 + (-96*u^3 + 2352*u^2 - 10496*u) * v + 4096.
G.f. (1/q) * (Product_{k>0} (1 + q^(2k-1)))^24 = 64 * (G_n)^24 where q = e^(-Pi sqrt(n)) and G_n is a Ramanujan class invariant.
A007191(n) = -(-1)^n * a(n).
a(n) ~ exp(2*Pi*sqrt(n)) / (2 * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
From Peter Bala, Sep 25 2023: (Start)
Laurent series g.f.: A(q) = - 16*lambda(-q)/lambda(q)^2 = 1/q + 24 + 276*q + 2048*q^2 + ..., where lambda(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... is the elliptic modular function in powers of the nome q = exp(i*Pi*t), the g.f. of A115977; lambda(q) = k(q)^2, where k(q) = (theta_2(q) / theta_3(q))^2 is the elliptic modulus.
A(q) = -16*(1 - lambda(-q))^2/lambda(-q). (End)
EXAMPLE
G.f. = 1/q + 24 + 276*q + 2048*q^2 + 11202*q^3 + 49152*q^4 + 184024*q^5 + ...
MATHEMATICA
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ ((1 - m) m / 16)^-1, {q, 0, n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[1 + q^k, {k, 1, n + 1, 2}]^24 / q, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^24 / q, {q, 0, n}]; (* Michael Somos, Nov 04 2014 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^24, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x^n * O(x); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^24, n))};
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Michael Somos, Aug 05 2004
STATUS
approved