A112216 McKay-Thompson series of class 92A 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

Offset

-1,9

Comments

Also McKay-Thompson series of class 92B for Monster. - Michel Marcus, Feb 19 2014

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

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

Index entries for McKay-Thompson series for Monster simple group

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of -1 + 1/q * chi(q) * chi(q^23) in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Aug 11 2015

Expansion of -1 + (eta(q^2) * eta(q^46))^2 / (eta(q) * eta(q^4) * eta(q^23) * eta(q^92)) in powers of q. - Michael Somos, Aug 11 2015

G.f.: -1 + 1/x * Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(46*k - 23)). - Michael Somos, Aug 11 2015

a(n) = A225956(n) unless n=0. a(2*n) = A092833(n). - Michael Somos, Aug 11 2015

a(n) ~ exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

Example

T92A = 1/q + q^2 + q^3 + q^4 + q^5 + q^6 + 2*q^7 + 2*q^8 + 2*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ -1 + 1/q QPochhammer[ -q, q^2] QPochhammer[ -q^23, q^46], {q, 0, n}]; (* Michael Somos, Aug 11 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( -x + (eta(x^2 + A) * eta(x^46 + A))^2 / (eta(x + A) * eta(x^4 + A) * eta(x^23 + A) * eta(x^92 + A)), n))}; /* Michael Somos, Aug 11 2015 */

Crossrefs

Cf. A092833, A225956.

Sequence in context: A213419 A000700 A081362 * A225956 A058688 A132322

Adjacent sequences: A112213 A112214 A112215 * A112217 A112218 A112219

Keyword

nonn

Author

Michael Somos, Aug 28 2005

Status

approved