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A007191
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McKay-Thompson series of class 2B for the Monster group with a(0) = -24.
(Formerly M5157)
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10
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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; internal format)
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OFFSET
| -1,2
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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 (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)
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REFERENCES
| J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=-1..1000
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, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group
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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 (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
a(n) = -(-1)^n * A097340(n). A007246(n) = a(n) unless n = 0.
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EXAMPLE
| 1/q - 24 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 - ...
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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 *)
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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) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^24, n))}
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CROSSREFS
| A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
Cf. A161195, A161196 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 06 2009]
Sequence in context: A045854 A014809 * A097340 A001496 A055754 A035707
Adjacent sequences: A007188 A007189 A007190 * A007192 A007193 A007194
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KEYWORD
| sign,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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