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A135213 McKay-Thompson series of class 30G for the Monster group with a(0) = -1. 2
1, -1, 1, -1, 2, -2, 2, -3, 5, -5, 5, -7, 9, -10, 11, -14, 18, -20, 22, -27, 32, -36, 40, -48, 57, -63, 70, -82, 95, -106, 119, -137, 158, -175, 195, -222, 252, -280, 311, -352, 397, -439, 486, -546, 611, -676, 747, -834, 929, -1024, 1128, -1253, 1389, -1528 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,5

COMMENTS

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

LINKS

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

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 q^(-1) * psi(q^3) * psi(q^5) / (psi(q) * psi(q^15)) in powers of q where psi() is a Ramanujan theta function.

Expansion of eta(q) * eta(q^6)^2 * eta(q^10)^2 * eta(q^15) / (eta(q^2)^2 * eta(q^3) * eta(q^5) * eta(q^30)^2) in powers of q.

Euler transform of period 30 sequence [-1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, 1, 0, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, 1, -1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + v) * (v - u^2) - 2 * u * (v - u).

G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A103259.

G.f.: 1 / ( x * Product_{k>0} P(15,x^k) * P(30,x^k)^2 ) where P(n,x) is the n-th cyclotomic polynomial.

a(n) = A058618(n) = A133098(n) unless n=0. Convolution inverse of A131794.

a(2*n) = - A094023(n). - Michael Somos, Oct 15 2015

a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

EXAMPLE

G.f. = 1/q - 1 + q - q^2 + 2*q^3 - 2*q^4 + 2*q^5 - 3*q^6 + 5*q^7 - 5*q^8 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ SeriesCoefficient[ EllipticTheta[ 2, 0, q^(3/2)] EllipticTheta[ 2, 0, q^(5/2)] / (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(15/2)]), {q, 0, n}]; (* Michael Somos, Oct 15 2015 *)

eta[q_]:= q^(1/24)*QPochhammer[q]; A:= eta[q]*eta[q^6]^2*eta[q^10]^2 *eta[q^15]/(eta[q^2]^2*eta[q^3]*eta[q^5]*eta[q^30]^2); a := CoefficientList[Series[q*A , {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 17 2018 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 * eta(x^10 + A)^2 * eta(x^15 + A) / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A)^2), n))};

CROSSREFS

Cf. A058618, A094023, A103259, A131794, A133098.

Sequence in context: A091974 A029073 A058618 * A145725 A058727 A304683

Adjacent sequences:  A135210 A135211 A135212 * A135214 A135215 A135216

KEYWORD

sign

AUTHOR

Michael Somos, Nov 23 2007

STATUS

approved

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Last modified January 18 13:05 EST 2019. Contains 319271 sequences. (Running on oeis4.)