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A007242 McKay-Thompson series of class 2a for the Monster group.
(Formerly M5455)
35
1, -492, -22590, -367400, -3764865, -28951452, -182474434, -990473160, -4780921725, -20974230680, -84963769662, -321583404672, -1147744866180, -3890805976500, -12601590210180, -39183052547592, -117437602167291, -340431109329600, -957251463332600, -2617490612355240, -6975126788952456, -18149106017123576, -46187557595906250 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A more correct name would be: Expansion of replicable function of class 2a. See Alexander et al., 1992. - N. J. A. Sloane, Jun 12 2015

From "More on Replicable Functions": 'The fifth row consists of the class names. As stated above, the numbers are the replication orders. For those functions arising in Monstrous Moonshine, the letter corresponds to the relevant conjugacy class in the Monster in Atlas notation (or, if there is more than one class, the one with the first letter). For non-monstrous functions, the class names use lower case letters and, in accordance with Atlas notation, are arranged generally in descending order of Frobenian.'

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

REFERENCES

T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see p. 425.

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..5000 (terms 0..500 from G. A. Edgar)

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

D. Alexander, C. Cummins, J. McKay and C. Simons, Completely replicable functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.

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

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.

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

FORMULA

Sqrt(j-1728), where j is the j-function, see A000521.

A014708(2*n - 1) == a(n) (mod 256). That is, the coefficients of (T1A - T2a) are all divisible by 256. - Michael Somos, Jun 29 2011

Expansion of (-phi(-q)^12 - 30 * phi(-q)^8 * phi(q)^4 + 96 * phi(-q)^4 * phi(q)^8 - 64 * phi(q)^12) / f(-q)^12 where phi(), f() are Ramanujan theta functions. - Michael Somos, Mar 17 2013

Expansion of (-8*(2*theta_2(0, q)^12-3*theta_2(0, q)^8*theta_3(0, q)^4-3*theta_3(0, q)^8*theta_2(0, q)^4+2*theta_3(0, q)^12))/(theta_3(0, q)^4*(theta_2(0, q)^4-theta_3(0, q)^4)*theta_2(0, q)^4) in powers of q.  Shows an analytic choice of the square root for complex q, 0 < |q| < 1. - G. A. Edgar, Mar 10 2017

G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/2). - Seiichi Manyama, Jul 02 2017

a(n) ~ -exp(2*Pi*sqrt(2*n)) / (2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jul 09 2017

EXAMPLE

T2a = 1/q - 492*q - 22590*q^3 - 367400*q^5 - 3764865*q^7 - ...

196884 - (-492) = 197376 = 256 * 771, 21493760 - 0 = 256 * 83960, ...

MATHEMATICA

a[ n_] :=  If[ n < 1, Boole[n == 0], SeriesCoefficient[ Sqrt[ 1728 (KleinInvariantJ[ Log[x] /(Pi I)] - 1) + O[x]^(2 n)], {x, 0, 2 n - 1}]] (* Michael Somos, Jun 29 2011 *)

nmax = 30; CoefficientList[Series[x^(1/2)*(-8*(2*EllipticTheta[2, 0, Sqrt[x]]^12 - 3*EllipticTheta[2, 0, Sqrt[x]]^8* EllipticTheta[3, 0, Sqrt[x]]^4 - 3*EllipticTheta[3, 0, Sqrt[x]]^8* EllipticTheta[2, 0, Sqrt[x]]^4 + 2*EllipticTheta[3, 0, Sqrt[x]]^12))/(EllipticTheta[3, 0, Sqrt[x]]^4*(EllipticTheta[2, 0, Sqrt[x]]^4 - EllipticTheta[3, 0, Sqrt[x]]^4)* EllipticTheta[2, 0, Sqrt[x]]^4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2017, check of formula by G. A. Edgar *)

eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 55; f1A := (eta[q]/eta[q^2] )^24*(1 +256*(eta[q^2]/eta[q])^24)^3; A007242:= CoefficientList[ Series[(q*f1A - 1728*q + O[q]^nmax)^(1/2), {q, 0, 50}], q]; Table[ A007242[[n]], {n, 1, 50}] (* G. C. Greubel, May 09 2018 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( ellj( x^2 * (1 + x * O(x^(2*n)) ) ) - 1728), 2*n - 1))} /* Michael Somos, Jun 29 2011 */

(PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, n, -504 * sigma(k, 5) * x^k, 1 + x * O(x^n)) / eta(x + x * O(x^n))^12, n))} /* Michael Somos, Mar 17 2013 */

CROSSREFS

(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), this sequence (k=12), A289063 (k=24).

Cf. A000521, A014708, A289061.

Sequence in context: A234672 A234667 A231455 * A198531 A300058 A251540

Adjacent sequences:  A007239 A007240 A007241 * A007243 A007244 A007245

KEYWORD

sign

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 19 17:45 EST 2019. Contains 319309 sequences. (Running on oeis4.)