A062244 McKay-Thompson series of class 36B for the Monster group.

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, -478, 98

Offset

0,8

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for sequences related to groups

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of a Hauptmodul for Gamma'_0(18).

G.f.: Product_{k>0} (1 + x^(6*k - 3))^3 / (1 + x^(2*k - 1)). - Michael Somos, Mar 17 2004

Expansion of q^(1/3) * eta(q) * eta(q^4) * eta(q^6)^6 / (eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3) in powers of q.

Given g.f. A(x), then B(q) = A(q^3) /q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v^4 - u^3*v^3 - 3*u^2*v^2 + v*u^4 + 4*u*v - 2. - Michael Somos, Mar 17 2004

Expansion of chi(x^3)^3 / chi(x) = f(-x, x^2) / psi(-x^3) = phi(x^3) / f(x, x^5) in powers of x where phi(), psi(), chi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Aug 10 2017

Euler transform of period 12 sequence [-1, 1, 2, 0, -1, -2, -1, 0, 2, 1, -1, 0, ...]. - Michael Somos, Sep 17 2007

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132972.

a(n) = (-1)^n * A062242(n). a(2*n) = A132179(n). a(2*n + 1) = - A092848(n).

Convolution inverse of A128111.

Example

G.f. = 1 - x + x^2 + x^3 - x^4 + x^6 - 2*x^7 + 2*x^9 - 3*x^10 + x^11 + ...
T36B = 1/q - q^2 + q^5 + q^8 - q^11 + q^17 - 2*q^20 + 2*q^26 - 3*q^29 + ...

Mathematica

a[ n_] := (-1)^n SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3]^3 / (QPochhammer[ x] QPochhammer[ x^6]^3), {x, 0, n}]; (* Michael Somos, Aug 20 2014 *)
A062244[n_] := SeriesCoefficient[QPochhammer[-q^3, q^6]^3/QPochhammer[-q, q^2], {q, 0, n}]; Table[A062244[n], {n, 0, 50}] (* G. C. Greubel, Aug 09 2017 *)
a[ n_] := SeriesCoefficient[ 2 x^(1/3) / (EllipticTheta[ 3, 0, x^(1/3)] / EllipticTheta[ 3, 0, x^3] - 1), {x, 0, n}]; (* Michael Somos, Aug 10 2017 *)
a[ n_] := SeriesCoefficient[ x^(1/3) (1 + EllipticTheta[2, Pi/4, x^(1/6)] / EllipticTheta[2, Pi/4, x^(3/2)]), {x, 0, n}]; (* Michael Somos, Aug 10 2017 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x^3]/(QPochhammer[ -x, x^6] QPochhammer[ -x^5, x^6] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Aug 10 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))}; /* Michael Somos, Jan 09 2005 */

Crossrefs

Cf. A062242, A092848, A128111, A132179, A132972.

Sequence in context: A328591 A210635 A062242 * A169979 A079957 A202036

Adjacent sequences: A062241 A062242 A062243 * A062245 A062246 A062247

Keyword

sign,easy

Author

N. J. A. Sloane, Jul 01 2001

Status

approved