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%I #69 Feb 21 2024 10:35:28
%S 1,24,276,2048,11202,49152,184024,614400,1881471,5373952,14478180,
%T 37122048,91231550,216072192,495248952,1102430208,2390434947,
%U 5061476352,10487167336,21301241856,42481784514,83300614144
%N McKay-Thompson series of class 4A for the Monster group with a(0) = 24.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C This series is also called Weber's modular function. - _N. J. A. Sloane_, Jun 23 2011
%C Or, better, it is the 24th power of Weber's modular function f(). - _Michael Somos_, Jan 10 2017
%C 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
%D 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.
%H T. D. Noe, <a href="/A097340/b097340.txt">Table of n, a(n) for n = -1..1000</a>
%H Barry Brent, <a href="https://www.doi.org/10.13140/RG.2.2.35825.45923">Finite field models of polynomials interpolating Fourier coefficients of modular functions for Hecke groups</a>, 2023.
%H Barry Brent, <a href="http://math.colgate.edu/~integers/y18/y18.pdf">Finite Field Models of Polynomials Interpolating Fourier Coefficients of Modular Functions for Hecke Groups</a>, Integers (2024) Vol. 24, Art. No. A18.
%H A. G. Greenhill, <a href="https://doi.org/10.1112/plms/s1-27.1.403">The Transformation and Division of Elliptic Functions</a>, Proceedings of the London Mathematical Society (1895) 403-486.
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.
%H Titus Piezas III, <a href="https://web.archive.org/web/20171109082029/http://www.oocities.org/titus_piezas/Pi_formulas1.pdf">Pi Formulas, Ramanujan, and the Baby Monster Group</a>
%H Titus Piezas III, <a href="http://www.oocities.org/titus_piezas/Ramanujan_a.pdf">Ramanujan's Constant exp(Pi sqrt(163)) And Its Cousins</a>
%H Titus Piezas III, <a href="https://sites.google.com/view/tpiezas/0011-article-1-the-j-function">0011: Article 1 (The j-function) - A collection of Algebraic Identities</a>
%H Titus Piezas III, <a href="https://sites.google.com/view/tpiezas/0022-part-1-the-163-dimensions">0022: The 163 Dimensions - A collection of Algebraic Identities</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 q^(-1) * chi(q)^24 where chi() is a Ramanujan theta function.
%F Expansion of (eta(q^2)^2 / (eta(q) * eta(q^4)))^24 in powers of q.
%F Euler transform of period 4 sequence [ 24, -24, 24, 0, ...].
%F G.f. is Fourier series of a level 4 modular function. f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t).
%F 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.
%F 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.
%F A007191(n) = -(-1)^n * a(n).
%F a(n) ~ exp(2*Pi*sqrt(n)) / (2 * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015
%F From _Peter Bala_, Sep 25 2023: (Start)
%F 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.
%F A(q) = -16*(1 - lambda(-q))^2/lambda(-q). (End)
%e G.f. = 1/q + 24 + 276*q + 2048*q^2 + 11202*q^3 + 49152*q^4 + 184024*q^5 + ...
%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ ((1 - m) m / 16)^-1, {q, 0, n}]]; (* _Michael Somos_, Jul 11 2011 *)
%t a[ n_] := SeriesCoefficient[ Product[1 + q^k, {k, 1, n + 1, 2}]^24 / q, {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^24 / q, {q, 0, n}]; (* _Michael Somos_, Nov 04 2014 *)
%t nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^24, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2015 *)
%o (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))};
%Y A007191, A007246, A045479, A035099, A097340, A107080, A134786 are all essentially the same sequence.
%Y Cf. A000700, A115977.
%K nonn
%O -1,2
%A _Michael Somos_, Aug 05 2004
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