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

%I M5157 #79 Oct 29 2023 21:28:54

%S 1,-24,276,-2048,11202,-49152,184024,-614400,1881471,-5373952,

%T 14478180,-37122048,91231550,-216072192,495248952,-1102430208,

%U 2390434947,-5061476352,10487167336,-21301241856,42481784514,-83300614144

%N McKay-Thompson series of class 2B for the Monster group with a(0) = -24.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C 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

%C From _Gary W. Adamson_, Jun 06 2009: (Start)

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

%C and row sums of triangle A161196. (End)

%C 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

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

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

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

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

%D 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.

%D 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.

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

%H Seiichi Manyama, <a href="/A007191/b007191.txt">Table of n, a(n) for n = -1..5000</a> (first 1001 terms from T. D. Noe)

%H R. E. Borcherds, <a href="http://www.math.berkeley.edu/~reb/papers/">Introduction to the monster Lie algebra</a>, 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.

%H B. Brent, <a href="http://www.emis.de/journals/EM/expmath/volumes/7/7.html">Quadratic Minima and Modular Forms</a>, Experimental Mathematics, v.7 no.3, 257-274.

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (<a href="http://www.math.ksu.edu/~gerald/papers/dr.pdf">pdf</a>, <a href="http://www.math.ksu.edu/~gerald/papers/dr.ps.gz">ps</a>).

%H Michael Somos, <a href="/A007191/a007191.pdf">Emails to N. J. A. Sloane, 1993</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

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

%F 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.

%F (eta(q)/eta(q^2))^24. - _Gene Ward Smith_, Aug 04 2006

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

%F Euler transform of period 2 sequence [-24, 0, ...]. - _Michael Somos_, Aug 19 2007

%F 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

%F 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

%F 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

%F 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

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

%F Convolution inverse of A014103.

%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)) / (2 * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015

%F 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

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

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^24 / q, {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)

%t a[ n_] := SeriesCoefficient[ Product[ 1 - q^k, {k, 1, n + 1, 2}]^24 / q, {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)

%t a[ n_] := With[ {m = ModularLambda[ Log[q] / (Pi I)]}, SeriesCoefficient[ (1 - m) / (m/16)^2, {q, 0, 2 n}]]; (* _Michael Somos_, Jul 11 2011 *)

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

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

%o (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))};

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

%Y Cf. A161195, A161196, A014103, A115977.

%K sign,easy,nice

%O -1,2

%A _N. J. A. Sloane_