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A097340 McKay-Thompson series of class 4A for the Monster group with a(0) = 24. 9
1, 24, 276, 2048, 11202, 49152, 184024, 614400, 1881471, 5373952, 14478180, 37122048, 91231550, 216072192, 495248952, 1102430208, 2390434947, 5061476352, 10487167336, 21301241856, 42481784514, 83300614144 (list; graph; refs; listen; history; text; internal format)
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

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

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], Sep 30 2015

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

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

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) = local(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.

Sequence in context: A045854 A014809 A007191 * A222156 A001496 A055754

Adjacent sequences:  A097337 A097338 A097339 * A097341 A097342 A097343

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 05 2004

STATUS

approved

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Last modified December 7 11:42 EST 2016. Contains 278874 sequences.