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McKay-Thompson series of class 12c for the Monster group with a(0) = 4.
3

%I #29 Feb 16 2025 08:33:14

%S 1,4,5,0,-5,0,9,0,-14,0,19,0,-34,0,55,0,-69,0,104,0,-164,0,209,0,-283,

%T 0,413,0,-539,0,712,0,-968,0,1248,0,-1642,0,2167,0,-2731,0,3526,0,

%U -4592,0,5736,0,-7244,0,9255,0,-11520,0,14378,0,-18018,0,22238,0

%N McKay-Thompson series of class 12c for the Monster group with a(0) = 4.

%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 Seiichi Manyama, <a href="/A187045/b187045.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..1001 from G. A. Edgar)

%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 J. M. Borwein and P. B. Borwein, <a href="http://dx.doi.org/10.1090/S0002-9947-1991-1010408-0">A cubic counterpart of Jacobi's identity and the AGM</a>, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.

%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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of (c(q) * b(q^2)^3) / (b(q) * b(q^4)^2 * c(q^4)) in powers of q where b(), c() are cubic AGM functions.

%F Expansion of (1/q) * chi(q)^5 * chi(-q) * chi(q^3) * chi(-q^3)^5 in powers of q where chi() is a Ramanujan theta function.

%F Expansion of (eta(q^2)^9 * eta(q^3)^4) / (eta(q)^4 * eta(q^4)^5 * eta(q^6)^3 * eta(q^12)) in powers of q.

%F Euler transform of period 12 sequence [ 4, -5, 0, 0, 4, -6, 4, 0, 0, -5, 4, 0, ...].

%F a(2*n) = 0 unless n=0. a(n) = A186930(n) unless n=0. a(2*n - 1) = A058491(n).

%e G.f. = 1/q + 4 + 5*q - 5*q^3 + 9*q^5 - 14*q^7 + 19*q^9 - 34*q^11 + 55*q^13 + ...

%t a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q, q^2] QPochhammer[ -q^3, q^6] (QPochhammer[ -q, q^2] QPochhammer[ q^3, q^6])^5, {q, 0, n}]; (* _Michael Somos_, Sep 05 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 * eta(x^3 + A)^4) / (eta(x + A)^4 * eta(x^4 + A)^5 * eta(x^6 + A)^3 * eta(x^12 + A)), n))};

%Y Cf. A058491, A186930.

%K sign,changed

%O -1,2

%A _Michael Somos_, Mar 07 2011