A058728 McKay-Thompson series of class 60D for the Monster group.

1, 0, -1, 1, 0, 0, 0, -1, 1, 1, -1, -1, 1, 0, -1, 2, 0, -2, 2, -1, 0, 2, -4, 0, 5, -1, -4, 2, 1, -2, 3, -3, -2, 7, -5, -2, 8, -6, -5, 8, 1, -5, 2, -2, -1, 12, -11, -10, 21, -6, -10, 13, -7, -4, 11, -7, -4, 14, -13, -10, 33, -14, -28, 32, -3, -12, 18, -24, 1, 36, -27, -22, 44, -13, -35, 50, -13, -36, 46, -26, -6, 56, -63, -22, 89, -30

Offset

-1,16

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

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

Michael Somos, A Remarkable eta-product Identity

Michael 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 1/q * (chi(-q^2) * chi(-q^30)) / (chi(-q^3) * chi(-q^5)) in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Feb 13 2017

Euler transform of a period 60 sequence. - Michael Somos, Feb 13 2017

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A145933. - Michael Somos, Feb 13 2017

G.f.: 1/x * Product_{k>0} (1 + x^(3*k)) * (1 + x^(5*k)) / ((1 + x^(2*k)) * (1 + x^(30*k))). - Michael Somos, Feb 13 2017

Convolution inverse of A132967. - Michael Somos, Feb 13 2017

Expansion of eta(q^2)*eta(q^6)*eta(q^10)*eta(q^30)/(eta(q^3)*eta(q^4)* eta(q^5)*eta(q^60)) in powers of q. - G. C. Greubel, Jun 06 2018

Example

T60D = 1/q - q + q^2 - q^6 + q^7 + q^8 - q^9 - q^10 + q^11 - q^13 + 2*q^14 - ...

Mathematica

QP = QPochhammer; s = q + QP[q]*QP[q^12]*QP[q^15]*(QP[q^20]/(QP[q^3]* QP[q^4]*QP[q^5]*QP[q^60])) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from A143751 *)
a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q^2, q^4] QPochhammer[ q^30, q^60] / (QPochhammer[ q^3, q^46] QPochhammer[ q^5, q^10]), {q, 0, n}]; (* Michael Somos, Feb 13 2017 *)

Prog

(PARI) {a(n) = my(A); n++; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A) / (eta(x^3 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^60 + A)), n))}; /* Michael Somos, Feb 13 2017 */

Crossrefs

Cf. A143751, A000521, A007240, A014708, A007241, A007267, A045478, etc.

Cf. A132967, A145933.

Sequence in context: A256232 A099751 A159937 * A143751 A353322 A158950

Adjacent sequences: A058725 A058726 A058727 * A058729 A058730 A058731

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2000

Extensions

More terms from N. J. A. Sloane, Sep 23 2009

Status

approved