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A138688 McKay-Thompson series of class 24I for the Monster group with a(0) = 2. 2
1, 2, 4, 6, 11, 18, 28, 42, 62, 90, 128, 180, 250, 342, 464, 624, 831, 1098, 1440, 1878, 2432, 3132, 4012, 5112, 6485, 8190, 10300, 12900, 16097, 20016, 24804, 30636, 37724, 46314, 56700, 69228, 84302, 102402, 124088, 150024, 180973, 217836, 261664, 313680 (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).

Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

LINKS

G. C. Greubel, Table of n, a(n) for n = -1..1000

K. Bringmann and H. Swisher, On a conjecture of Koike on identities between Thompson series and Roger-Ramanujan functions, Proc. Amer. Math. Soc. 135 (2007), 2317-2326. See page 2325 (A.7).

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

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

Expansion of psi(q^4) * phi(-q^3) / (phi(-q) * psi(q^12)) in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of eta(q^2) * eta(q^3)^2 * eta(q^8)^2 * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6) * eta(q^24)^2) in powers of q.

Euler transform of period 24 sequence [ 2, 1, 0, 2, 2, 0, 2, 0, 0, 1, 2, 0, 2, 1, 0, 0, 2, 0, 2, 2, 0, 1, 2, 0, ...].

G.f.: (G(x) * G(x^24) + x^5 * H(x) * H(x^24))^2 * (G(x^4) * G(x^6) + x^2 * H(x^4) * H(x^6)) where G() and H() are Rogers-Ramanujan functions.

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058579(n) unless n=0.

a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(5/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

EXAMPLE

G.f. = 1/q + 2 + 4*q + 6*q^2 + 11*q^3 + 18*q^4 + 28*q^5 + 42*q^6 + 62*q^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^2] EllipticTheta[ 4, 0, q^3] / (EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^6]), {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)

nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(3*k))^2 * (1-x^(4*k)) * (1+x^(4*k))^2 * (1+x^(6*k)) / ((1-x^k) * (1-x^(24*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^8 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)^2), n))};

CROSSREFS

Cf. A058579.

Sequence in context: A034297 A026636 A026658 * A131298 A168445 A185192

Adjacent sequences:  A138685 A138686 A138687 * A138689 A138690 A138691

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 26 2008

STATUS

approved

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Last modified January 22 18:06 EST 2019. Contains 319365 sequences. (Running on oeis4.)