-1,2

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 (genewardsmith(AT)gmail.com), Aug 04 2006

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 06 2009: (Start)

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

and row sums of triangle A161196 (End)

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

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

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

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.

T. D. Noe, Table of n, a(n) for n=-1..1000

Index entries for McKay-Thompson series for Monster simple group

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.

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

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

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 Ramanujan's class invariant.

(eta(q)/eta(q^2))^24 - Gene Ward Smith (genewardsmith(AT)gmail.com), 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 a modular elliptic function. - Michael Somos Aug 19 2007

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 Fourier series which satisfies f(-1/(2 t)) = 4096 / f(t) where q = exp(2 pi i t). - Michael Somos Aug 19 2007

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

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

Apart from constant term, same as A035099, A007246 and A045479.

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

A161195, A161196 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 06 2009]

Sequence in context: A010940 A045854 A014809 this_sequence A097340 A001496 A055754

Adjacent sequences: A007188 A007189 A007190 this_sequence A007192 A007193 A007194

sign,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).