%I #33 Mar 12 2021 22:24:42

%S 1,5,-5,9,-14,19,-34,55,-69,104,-164,209,-283,413,-539,712,-968,1248,

%T -1642,2167,-2731,3526,-4592,5736,-7244,9255,-11520,14378,-18018,

%U 22238,-27556,34132,-41701,51184,-62900,76323,-92771,113002,-136421,164673,-198842

%N Coefficients of replicable function number 12c.

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

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

%H G. C. Greubel, <a href="/A058491/b058491.txt">Table of n, a(n) for n = 0..1000</a>

%H D. Alexander, C. Cummins, J. McKay and C. Simons, <a href="http://oeis.org/A007242/a007242_1.pdf">Completely Replicable Functions</a>, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F 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

%F 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

%F a(n) = A186930(2*n - 1) = A187045(2*n - 1). - _Michael Somos_, Aug 20 2012

%e 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 + ...

%e T12c = 1/q + 5*q - 5*q^3 + 9*q^5 - 14*q^7 + 19*q^9 - 34*q^11 + 55*q^13 - ...

%t 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 *)

%o (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 */

%o (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

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

%Y Cf. A186930, A187045.

%K sign

%O 0,2

%A _N. J. A. Sloane_, Nov 27 2000

%E Name changed by _N. J. A. Sloane_, Jun 10 2015 at the suggestion of _Yang-Hui He_