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A112211 McKay-Thompson series of class 84B for the Monster group. 2
1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 1, -1, 1, 0, 1, -2, 2, -2, 1, 0, 1, -2, 3, -3, 1, 0, 2, -4, 5, -4, 2, -2, 3, -4, 6, -6, 3, -2, 3, -7, 10, -8, 5, -4, 5, -10, 13, -10, 6, -4, 7, -14, 16, -14, 11, -8, 11, -18, 22, -21, 14, -8, 14, -24, 29, -26, 17, -14, 22, -32, 39, -36, 24, -20, 28, -40, 49, -44, 32, -28, 34, -52, 67 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,18

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. 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 chi(-q) * chi(-q^21) / (chi(-q^3) * chi(-q^7)) in powers of q where chi() is a Ramanujan theta function.

Expansion of q^(1/2) * eta(q) * eta(q^6) * eta(q^14) * eta(q^21) / (eta(q^2) * eta(q^3) * eta(q^7) * eta(q^42)) in powers of q.

Given g.f. A(x), then B(x) = A(x^2) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = (1 + v) * (u^2*w^2 - v^2) - (v^2 - v) * (u^2 + w^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (168 t)) = 1 / f(t) where q = exp(2 Pi i t).

G.f.: Product_{k>0} (1 + x^(3*k)) * (1 + x^(7*k)) / ((1 + x^k) * (1 + x^(21*k))).

G.f.: Product_{k>0} 1 / P(42,x^k) where P(n,x) is the n-th cyclotomic polynomial.

a(n) ~ (-1)^n * exp(sqrt(2*n/21)*Pi) / (2^(5/4) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 28 2018

EXAMPLE

T84B = 1/q -q -q^9 +q^11 -q^21 +q^23 -q^25 +q^27 +q^31 -2*q^33 +...

MATHEMATICA

QP = QPochhammer; s = QP[q]*QP[q^6]*QP[q^14]*(QP[q^21]/(QP[q^2]*QP[q^3]* QP[q^7]*QP[q^42])) + O[q]^90; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 15 2015, from 2nd formula *)

PROG

(PARI) q='q+O('q^50); Vec(eta(q)*eta(q^6)*eta(q^14)*eta(q^21)/(eta(q^2)* eta(q^3)*eta(q^7)*eta(q^42))) \\ G. C. Greubel, Jun 20 2018

CROSSREFS

Convolution inverse of A109368.

Sequence in context: A072931 A307013 A307012 * A246575 A112215 A176389

Adjacent sequences:  A112208 A112209 A112210 * A112212 A112213 A112214

KEYWORD

sign

AUTHOR

Michael Somos, Aug 28 2005, Jan 12 2009

STATUS

approved

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Last modified October 18 18:56 EDT 2019. Contains 328197 sequences. (Running on oeis4.)