A058574 McKay-Thompson series of class 24D for the Monster group.

1, -1, -1, -1, 2, 1, -2, 1, 3, 0, -4, -1, 5, -1, -7, 0, 8, 0, -10, 1, 13, 2, -16, 0, 20, -3, -24, -2, 30, 2, -36, 4, 43, 0, -52, -3, 61, -2, -73, -1, 86, 1, -102, 3, 120, 4, -140, -1, 165, -8, -192, -5, 224, 6, -260, 10, 301, 2, -348, -7, 401, -7, -462, -2, 530, 2, -608

Offset

0,5

Comments

Ramanujan theta functions: f(q) := Product_{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) := Product_{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, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of chi(-x) * chi(-x^2) * chi(-x^3) * chi(-x^6) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Mar 06 2011

Expansion of q^(1/2) * eta(q) * eta(q^3) / (eta(q^4) * eta(q^12)) in powers of q. - Michael Somos, Mar 06 2011

Euler transform of period 12 sequence [ -1, -1, -2, 0, -1, -2, -1, 0, -2, -1, -1, 0, ...]. - Michael Somos, Mar 06 2011

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 4 / f(t) where q = exp(2 Pi i t). - Michael Somos, Mar 06 2011

Convolution square is A187196. a(n) = (-1)^n * A112165(n). - Michael Somos, Mar 06 2011

Example

T24D = 1/q - q - q^3 - q^5 + 2*q^7 + q^9 - 2*q^11 + q^13 + 3*q^15 - 4*q^19 + ...

Mathematica

QP = QPochhammer; s = QP[q]*(QP[q^3]/(QP[q^4]*QP[q^12])) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, from 2nd formula *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(1/2)* eta[q]*eta[q^3]/(eta[q^4]*eta[q^12]), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 14 2018 *)

Prog

(PARI) q='q+O('q^50); A=eta(q)*eta(q^3)/(eta(q^4)*eta(q^12)); Vec(A) \\ G. C. Greubel, Jun 14 2018

Crossrefs

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

Cf. A112165, A187196. - Michael Somos, Mar 06 2011

Sequence in context: A355341 A245714 A092953 * A112165 A112186 A112187

Adjacent sequences: A058571 A058572 A058573 * A058575 A058576 A058577

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2000

Status

approved