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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 (list; graph; refs; listen; history; text; internal format)
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.

M. Somos, Introduction to Ramanujan theta functions

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.

Index entries for sequences related to groups

Index entries for McKay-Thompson series for Monster simple group

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

Cf. A062244, A092848, A132179.

Sequence in context: A063514 A082490 A210635 * A062244 A169979 A079957

Adjacent sequences:  A062239 A062240 A062241 * A062243 A062244 A062245

KEYWORD

sign

AUTHOR

N. J. A. Sloane, Jun 30 2001

STATUS

approved

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)