A082304 McKay-Thompson series of class 16d for the Monster group.

1, -2, -1, 2, 3, -2, -4, 4, 5, -8, -8, 10, 11, -12, -15, 18, 22, -26, -29, 34, 38, -42, -51, 56, 66, -78, -85, 98, 109, -120, -139, 156, 176, -202, -222, 250, 279, -306, -346, 384, 429, -482, -530, 590, 650, -714, -797, 876, 972, -1080, -1180, 1304, 1431, -1562, -1728, 1892, 2078, -2290, -2496

Offset

0,2

Comments

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

Links

Seiichi Manyama, 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. see page 273.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of phi(-q) / psi(q^2) in powers of q where phi(), psi() are Ramanujan theta functions.

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

Euler transform of period 4 sequence [ -2, -2, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A001936. - Michael Somos, Jul 04 2014

Given g.f. A(x), then B(q) = A(q)^4 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u * (16 + u) * (16 + v). - Michael Somos, Jul 04 2014

Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - u*v * (4 + u*v)^2. - Michael Somos, Aug 13 2007

Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^5)) where f(u, v) = u*v * (16 + u^2*v^2)^2 - (u+v)^2 * (u^2 - 6*u*v + v^2)^2. - Michael Somos, Jul 04 2014

G.f.: Product_{k>0} ((1 - x^k) / (1 - x^(4*k)))^2.

a(n) = (-1)^n * A029839(n). Convolution inverse of A001936. - Michael Somos, Jul 04 2014

abs(a(n)) ~ exp(Pi*sqrt(n)/2) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Feb 07 2023

Example

T16d = 1/q - 2*q^3 - q^7 + 2*q^11 + 3*q^15 - 2*q^19 - 4*q^23 + 4*q^27 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^4])^2, {x, 0, n}]; (* Michael Somos, Jul 04 2014 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^4 + A))^2, n))};

Crossrefs

Cf. A001936, A029839.

Sequence in context: A096920 A087154 A029839 * A321664 A250099 A241949

Adjacent sequences: A082301 A082302 A082303 * A082305 A082306 A082307

Keyword

sign

Author

Michael Somos, Apr 08 2003

Status

approved