A097340 McKay-Thompson series of class 4A for the Monster group with a(0) = 24.

1, 24, 276, 2048, 11202, 49152, 184024, 614400, 1881471, 5373952, 14478180, 37122048, 91231550, 216072192, 495248952, 1102430208, 2390434947, 5061476352, 10487167336, 21301241856, 42481784514, 83300614144

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

A007191, A007246, A045479, A035099, A097340, A107080, A134786 are all essentially the same sequence.

Cf. A000700, A115977.

Sequence in context: A045854 A014809 A007191 * A222156 A297604 A001496

Adjacent sequences: A097337 A097338 A097339 * A097341 A097342 A097343

Keyword

nonn

Author

Michael Somos, Aug 05 2004

Status

approved