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A112216 McKay-Thompson series of class 92A for the Monster group. 3
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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

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Last modified October 18 20:21 EDT 2019. Contains 328197 sequences. (Running on oeis4.)