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A058530 McKay-Thompson series of class 17A for the Monster simple group. 3
1, 0, 7, 14, 29, 50, 92, 148, 246, 386, 603, 904, 1367, 1996, 2914, 4160, 5924, 8290, 11581, 15942, 21878, 29712, 40184, 53876, 71979, 95436, 126097, 165556, 216594, 281848, 365548, 471808, 607050, 777794, 993528, 1264338, 1604434, 2029026 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (17 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 06 2018

LINKS

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

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

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 A^2 - 2, where A = q^(1/2)*(eta(q^4)^2*(eta(q^34)^5 /(eta(q)*eta(q^2)*eta(q^17)^3*eta(q^68)^2)) - eta(q^2)^5*(eta(q^68)^2 /(eta(q)^3*eta(q^4)^2*eta(q^17)*eta(q^34)))), in powers of q. - G. C. Greubel, Jun 14 2018

a(n) ~ exp(4*Pi*sqrt(n/17)) / (sqrt(2) * 17^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 28 2018

Expansion of -2 + q^(-1) * ((psi(q^2) * phi(q^17) - q^4 * phi(q) * psi(q^34)) / (f(-q) * f(-q^17)))^2 in powers of q where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 06 2018

EXAMPLE

T17A = 1/q + 7*q + 14*q^2 + 29*q^3 + 50*q^4 + 92*q^5 + 148*q^6 + 246*q^7 + ...

MATHEMATICA

eta[q_]:= q^(1/24)*QPochhammer[q]; A:= q^(1/2)*(eta[q^4]^2*(eta[q^34]^5 /(eta[q]*eta[q^2]*eta[q^17]^3*eta[q^68]^2)) - eta[q^2]^5*(eta[q^68]^2 /(eta[q]^3*eta[q^4]^2*eta[q^17]*eta[q^34]))); a:= CoefficientList[ Series[A^2 - 2*q, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 14 2018 *)

a[ n_] := SeriesCoefficient[ -2 + ((EllipticTheta[ 2, 0, q] EllipticTheta[ 3, 0, q^17] - EllipticTheta[ 2, 0, q^17] EllipticTheta[ 3, 0, q]) / (QPochhammer[ q] QPochhammer[ q^16]))^2 / (4 q^(3/2)), {q, 0, n}]; (* Michael Somos, Sep 06 2018 *)

PROG

(PARI) q='q+O('q^50); A = (eta(q^4)^2*(eta(q^34)^5/(eta(q)*eta(q^2)* eta(q^17)^3*eta(q^68)^2)) - q^4*eta(q^2)^5*(eta(q^68)^2/(eta(q)^3* eta(q^4)^2*eta(q^17)*eta(q^34)))); Vec(A^2 - 2*q) \\ G. C. Greubel, Jun 14 2018

CROSSREFS

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Cf. A152944 (same sequence except for n=0).

Sequence in context: A005009 A135092 A245417 * A293359 A134384 A304143

Adjacent sequences:  A058527 A058528 A058529 * A058531 A058532 A058533

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 27 2000

EXTENSIONS

More terms from Michel Marcus, Feb 19 2014

STATUS

approved

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Last modified January 16 15:53 EST 2019. Contains 319195 sequences. (Running on oeis4.)