A058688 McKay-Thompson series of class 46A for the Monster group.

1, 0, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -10, 12, -12, 13, -15, 17, -18, 19, -22, 25, -27, 28, -32, 36, -38, 41, -46, 51, -54, 58, -64, 71, -76, 81, -89, 99, -105, 112, -123, 134, -143, 153, -167, 182, -194, 207, -225, 244, -260, 277, -301, 325, -346, 369, -398, 429, -458

Offset

-1,9

Comments

Also McKay-Thompson series of class 46B for the Monster group.

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..5000

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

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 + (1/q) * chi(-q) * chi(-q^23) in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Jun 07 2006

G.f.: 1 + (1/x) * Product_{k>0} 1 / ((1 + x^k) * (1 + x^(23*k))).

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - v + 2 - 2*u + u^2 - v*u^2. - Michael Somos, Feb 14 2007

a(n) = A132322(n) unless n=0.

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Expansion of 1 + eta(q)*eta(q^23)/(eta(q^2)*eta(q^46)) in powers of q. - G. C. Greubel, Jun 16 2018

Example

T46A = 1/q - q^2 + q^3 - q^4 + q^5 - q^6 + 2*q^7 - 2*q^8 + 2*q^9 + ...

Mathematica

QP = QPochhammer; s = q + QP[q]*(QP[q^23]/(QP[q^2]*QP[q^46])) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, adapted from PARI *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(1+eta[q] *eta[q^23]/(eta[q^2]*eta[q^46])), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 16 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( x + eta(x + A) * eta(x^23 + A) / (eta(x^2 + A) * eta(x^46 + A)), n))} /* Michael Somos, Feb 14 2007 */

Crossrefs

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

Cf. A132322.

Sequence in context: A081362 A112216 A225956 * A132322 A018118 A029084

Adjacent sequences: A058685 A058686 A058687 * A058689 A058690 A058691

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2000

Status

approved