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A082304 McKay-Thompson series of class 16d for the Monster group. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..58.

M. 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

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

Given g.f. A(x), then B(x)=A(x^4)/x satisfies 0=f(B(x), B(x^2)) where f(u, v)=-v^2 +16uv +16u^2 +256u +u^2v. - Michael Somos May 14 2004

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

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

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

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

EXAMPLE

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

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

A029839(n)= (-1)^n* a(n).

Sequence in context: A096920 A087154 A029839 * A214720 A035368 A107853

Adjacent sequences:  A082301 A082302 A082303 * A082305 A082306 A082307

KEYWORD

sign

AUTHOR

Michael Somos, Apr 08 2003

STATUS

approved

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Last modified April 16 14:51 EDT 2014. Contains 240600 sequences.