A058491 Coefficients of replicable function number 12c.

1, 5, -5, 9, -14, 19, -34, 55, -69, 104, -164, 209, -283, 413, -539, 712, -968, 1248, -1642, 2167, -2731, 3526, -4592, 5736, -7244, 9255, -11520, 14378, -18018, 22238, -27556, 34132, -41701, 51184, -62900, 76323, -92771, 113002, -136421, 164673, -198842

Offset

0,2

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

D. Alexander, C. Cummins, J. McKay and C. Simons, Completely Replicable Functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.

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 a(x) / (psi(x) * psi(x^3)) in powers of x where psi() is a Ramanujan theta function and a() is a cubic AGM theta function. - Michael Somos, Aug 20 2012

Expansion of q^(1/2) * (eta(q)^3 + 9 * q * eta(q^9)^3) * eta(q) / (eta(q^2) * eta(q^6))^2 in powers of q. - Michael Somos, Aug 20 2012

a(n) = A186930(2*n - 1) = A187045(2*n - 1). - Michael Somos, Aug 20 2012

Example

G.f. = 1 + 5*x - 5*x^2 + 9*x^3 - 14*x^4 + 19*x^5 - 34*x^6 + 55*x^7 - 69*x^8 + ...
T12c = 1/q + 5*q - 5*q^3 + 9*q^5 - 14*q^7 + 19*q^9 - 34*q^11 + 55*q^13 - ...

Mathematica

a[n_]:= SeriesCoefficient[(QPochhammer[x]^3 + 9*x*QPochhammer[x^9]^3)* QPochhammer[x]/(QPochhammer[x^2]*QPochhammer[x^6])^2, {x, 0, n}]; (* Michael Somos, Sep 14 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) * eta(x + A) / (eta(x^2 + A) * eta(x^6 + A))^2, n))}; /* Michael Somos, Aug 20 2012 */
(PARI) { my(q='q+O('q^66)); Vec( (eta(q)^3 + 9 * q * eta(q^9)^3) * eta(q) / (eta(q^2) * eta(q^6))^2 ) } \\ Joerg Arndt, Apr 16 2017

Crossrefs

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

Cf. A186930, A187045.

Sequence in context: A290968 A011986 A047880 * A058584 A324792 A147197

Adjacent sequences: A058488 A058489 A058490 * A058492 A058493 A058494

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2000

Extensions

Name changed by N. J. A. Sloane, Jun 10 2015 at the suggestion of Yang-Hui He

Status

approved