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A062242
McKay-Thompson series of class 18D for the Monster group.
14
1, 1, 1, -1, -1, 0, 1, 2, 0, -2, -3, -1, 4, 4, 1, -4, -6, -1, 5, 8, 1, -8, -10, -2, 11, 14, 4, -14, -19, -4, 17, 24, 4, -23, -31, -6, 31, 40, 9, -38, -50, -10, 46, 63, 11, -60, -79, -16, 77, 98, 21, -92, -122, -24, 112, 150, 28, -140, -183, -36, 173, 224, 46, -208, -273, -54, 249, 329, 62, -304, -396, -78, 370
OFFSET
0,8
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 7 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014
There is a typo in the entry for this q-series in Table I of Yang 2004. The exponent of 18 should be 3. - Michael Somos, Jul 21 2014
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(18). [Yang 2004] - Michael Somos, Jul 21 2014
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
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
G.f.: Product_{k>0} (1 - x^(6*k - 3))^3 / (1 - x^(2*k-1)). - Michael Somos, Mar 17 2004
Expansion of chi(-q^3)^3 / chi(-q) in powers of q where chi() is a Ramanujan theta function.
Expansion of q^(1/3) * c(q) / c(q^2) in powers of q where c() is a cubic AGM theta function. - Michael Somos, Oct 17 2006
Expansion of q^(1/3) * eta(q^2) * eta(q^3)^3 / (eta(q) * eta(q^6)^3) in powers of q. - Michael Somos, Mar 05 2004
Euler transform of period 6 sequence [ 1, 0, -2, 0, 1, 0, ...]. - Michael Somos, Mar 05 2004
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 2*u +v^2 - u^2*v. - Michael Somos, Mar 17 2004
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - v + v^2) * u^3 - (4 + 2*v + v^2) * v. - Michael Somos, Aug 11 2007
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^5)) where f(u, v) = u^6 + v^6 - u^5*v^5 + 5*u^4*v^4 - 20*u^3*v^3 + 20*u^2*v^2 - 16*u*v + 5*u^2*v^5 + 5*u^5*v^2 - 10*u^4*v - 10*u*v^4. - Michael Somos, Aug 11 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128128.
G.f.: 1 + x*(1+x)/(1 + x^2*(1+x^2)/(1 + x^3*(1+x^3)/(1 + x^4*(1+x^4)/(1 + x^5*(1+x^5)/(1 + ...))))), a continued fraction. - Paul D. Hanna, Jul 09 2013
a(n) = (-1)^n * A062244(n). a(2*n) = A132179(n). a(2*n + 1) = A092848(n).
Convolution inverse of A092848.
EXAMPLE
G.f. = 1 + x + x^2 - x^3 - x^4 + x^6 + 2*x^7 - 2*x^9 - 3*x^10 - x^11 + 4*x^12 + ...
T18D = 1/q + q^2 + q^5 - q^8 - q^11 + q^17 + 2*q^20 - 2*q^26 - 3*q^29 - x^32 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3]^3 / (QPochhammer[ x] QPochhammer[ x^6]^3), {x, 0, n}]; (* Michael Somos, Jul 21 2014 *)
PROG
(PARI) {a(n) = local(A, m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=3; A = subst(A, x, x^3) / x; A = (x*A * (4 + 2*A + A^2) / (1 - A + A^2))^(1/3)); polcoeff(A, n))};
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3), n))};
(PARI) /* Continued Fraction Expansion: */
{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1 + x^(n-k+1)*(1 + x^(n-k+1))/(CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 09 2013
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Jun 30 2001
STATUS
approved