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A058487 McKay-Thompson series of class 12I for the Monster group. 10
1, 2, 1, 0, -2, -2, 2, 4, 3, -4, -8, -4, 5, 14, 7, -8, -20, -12, 14, 28, 17, -20, -44, -24, 28, 66, 36, -40, -90, -52, 56, 124, 71, -80, -176, -96, 109, 244, 133, -144, -326, -182, 198, 432, 240, -268, -580, -316, 349, 772, 420, -456, -1004, -552, 600, 1300, 713, -780, -1692, -916, 1001, 2186, 1182 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 5 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(12). [Yang 2004] - Michael Somos, Jul 21 2014
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000 (terms 0..502 from G. A. Edgar)
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
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 (psi(x) / psi(x^3))^2 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jul 21 2014
G.f.: ( Product_{k>0} (1 - x^(6*k - 2)) * (1 - x^(6*k - 4)) / ((1 - x^(6*k - 1)) * (1 - x^(6*k - 5))) )^2.
Expansion of q^(1/2) * (eta(q^2)^4 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)^4)) in powers of q.
Euler transform of period 6 sequence [ 2, -2, 0, -2, 2, 0,...]. - Michael Somos, Mar 18 2004
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 + 3*v - u^2*v + v^2. - Michael Somos, Mar 18 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A186924.
a(n) = (-1)^n * A062243(n).
Convolution square is A128633. Convolution inverse of A217786. - Michael Somos, Jul 21 2014
EXAMPLE
G.f. = 1 + 2*x + x^2 - 2*x^4 - 2*x^5 + 2*x^6 + 4*x^7 + 3*x^8 - 4*x^9 - 8*x^10 + ...
T12I = 1/q + 2*q + 1*q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 - 4*q^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^2 / EllipticTheta[ 2, 0, q^3]^2, {q, 0, 2 n - 1}]; (* Michael Somos, Jul 21 2014 *)
QP = QPochhammer; s = QP[q^2]^4*(QP[q^3]^2/(QP[q]^2*QP[q^6]^4)) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015 *)
PROG
(PARI) {a(n) = local(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A * (A + 3*x) / (A - x))); polcoeff(A, n))};
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2))^2, n))};
CROSSREFS
Sequence in context: A047654 A358477 A351981 * A062243 A128095 A316658
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Nov 27 2000
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)