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A007191 McKay-Thompson series of class 2B for the Monster group with a(0) = -24.
(Formerly M5157)
18
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).

Let t(q) = (eta(q) / eta(q^2))^24 = 1/q - 24 + 276q - 2048q^2 + ... If j(q) is the q-series for the j-invariant, with coefficients from A000521, then j(q) = (t + 256)^3/t^2 j(q^2) = (t + 16)^3/t. Hence t can be used to parametrize the classical modular curve X0(2). - Gene Ward Smith, Aug 04 2006

From Gary W. Adamson, Jun 06 2009: (Start)

Equals (1/q) * the convolution square of A161195: (1, -12, 66, -232, 639, ...)

and row sums of triangle A161196. (End)

Given g.f. A(q), Greenhill (1895) denotes -1/64 * A(q) by tau_oo on page 409 equation (43). - Michael Somos, Jul 17 2013

REFERENCES

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 371. Eq. (1)

A. G. Greenhill, The Transformation and Division of Elliptic Functions, Proceedings of the London Mathematical Society (1895) 403-486.

G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18 (1990), no. 1, 253-278.

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe and Seiichi Manyama, Table of n, a(n) for n = -1..5000 (first 1001 terms from T. D. Noe)

R. E. Borcherds, Introduction to the monster Lie algebra, pp. 99-107 of M. Liebeck and J. Saxl, editors, Groups, Combinatorics and Geometry (Durham, 1990). London Math. Soc. Lect. Notes 165, Cambridge Univ. Press, 1992.

B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).

M. Somos, Emails to N. J. A. Sloane, 1993

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for sequences related to groups

Index entries for McKay-Thompson series for Monster simple group

FORMULA

G.f.: (1/x)(Product_{k>0} 1/(1 + x^k))^24.

G.f.: (1/q)(Product_{k>0} (1 - q^(2*k - 1)))^24 = 64 * (g_n)^24 where q = e^(-Pi sqrt(n)) and g_n is Ramanujan's class invariant.

(eta(q)/eta(q^2))^24. - Gene Ward Smith, Aug 04 2006

Expansion of q^(-1) * chi(-q)^24 in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Aug 19 2007

Euler transform of period 2 sequence [-24, 0, ...]. - Michael Somos, Aug 19 2007

Expansion of (1 - lambda(t)) / (lambda(t) / 16)^2 in powers of q = exp(2 Pi i t) where lambda() is the elliptic modular function A115977. - Michael Somos, Aug 19 2007

Expansion of 64 tau(omega) in powers of q = exp(2 Pi i omega) where tau() is Fricke's function on page 371 equation (1). - Michael Somos, Jun 12 2012

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2*v - v^2 + 48*u*v + 4096*u. - Michael Somos, Aug 19 2007

G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 4096 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A014103. - Michael Somos, Aug 19 2007

a(n) = -(-1)^n * A097340(n). A007246(n) = a(n) unless n = 0.

Convolution inverse of A014103.

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)) / (2 * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015

a(-1) = 1, a(n) = -(24/(n+1))*Sum_{k=1..n+1} A000593(k)*a(n-k) for n > -1. - Seiichi Manyama, Mar 29 2017

EXAMPLE

G.f. = 1/q - 24 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 - ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^24 / q, {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_] := With[ {m = ModularLambda[ Log[q] / (Pi I)]}, SeriesCoefficient[ (1 - m) / (m/16)^2, {q, 0, 2 n}]]; (* Michael Somos, Jul 11 2011 *)

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m) / (m/16)^2, {q, 0, 2 n}]]; (* Michael Somos, Jul 11 2011 *)

PROG

(PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^-24, n))};

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^24, n))};

CROSSREFS

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

Cf. A161195, A161196, A014103, A115977.

Sequence in context: A010940 A045854 A014809 * A097340 A222156 A297604

Adjacent sequences:  A007188 A007189 A007190 * A007192 A007193 A007194

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 21 08:47 EDT 2019. Contains 328292 sequences. (Running on oeis4.)