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A105559 McKay-Thompson series of class 6E for the Monster group with a(0) = 3. 8
1, 3, 6, 4, -3, -12, -8, 12, 30, 20, -30, -72, -46, 60, 156, 96, -117, -300, -188, 228, 552, 344, -420, -1008, -603, 732, 1770, 1048, -1245, -2976, -1776, 2088, 4908, 2900, -3420, -7992, -4658, 5460, 12756, 7408, -8583, -19944, -11564, 13344, 30756, 17744, -20448, -46944 (list; graph; refs; listen; history; text; internal format)
OFFSET
-1,2
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).
Number 2 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(6). [Yang 2004] - Michael Somos, Jul 21 2014
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..147 from G. A. Edgar)
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
FORMULA
Expansion of (eta(q^2) * eta(q^3)^3 / (eta(q) * eta(q^6)^3))^3 in powers of q.
G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = v^2 + 8*u + 6*u*v - u^2*v.
G.f.: x^-1 (Product_{k>0} (1 - x^(6*k - 3))^3 / (1 - x^(2*k - 1)))^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128643.
Expansion of (c(q) / c(q^2))^3 in powers of q where c() is a cubic AGM theta function.
Expansion of q^(-1) * (chi(-q^3)^3 / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.
Euler transform of period 6 sequence [ 3, 0, -6, 0, 3, 0, ...].
a(n) = A007258(n) unless n=0. Convolution inverse of A123633.
Convolution cube of A062242. - Michael Somos, Apr 24 2015
EXAMPLE
G.f. = 1/q + 3 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3, q^6]^3 / QPochhammer[ q, q^2])^3 / q, {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)
a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 3, n, 6}] / Product[ 1 - q^k, {k, 1, n, 2}]^3)^3 / q, {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ( eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3) )^3, n))};
CROSSREFS
Sequence in context: A231737 A140072 A187148 * A090038 A308291 A006464
KEYWORD
sign
AUTHOR
Michael Somos, Apr 13 2005, Jan 21 2009
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)