A187197 McKay-Thompson series of class 12E for the Monster group with a(0) = 2.

1, 2, -1, 0, 7, 0, -9, 0, 10, 0, -23, 0, 38, 0, -47, 0, 75, 0, -112, 0, 148, 0, -217, 0, 293, 0, -385, 0, 553, 0, -728, 0, 928, 0, -1272, 0, 1670, 0, -2111, 0, 2765, 0, -3566, 0, 4504, 0, -5784, 0, 7300, 0, -9123, 0, 11592, 0, -14458, 0, 17838, 0, -22342, 0, 27668, 0, -33884

Offset

-1,2

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 = -1..1000

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1/q) * ((psi(q) * psi(q^3)) / (psi(q^2) * psi(q^6)))^2 in powers of q where psi() is a Ramanujan theta function.

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

Euler transform of period 12 sequence [ 2, -4, 4, 0, 2, -8, 2, 0, 4, -4, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123861.

Given g.f. A(q) then 0 = f(A(q), A(q^2)) where f(u, v) = (v - 4)^2 - u*v * (u - 4). - Michael Somos, Aug 31 2014

Convolution square of A112165. a(n) = A187196(n) unless n=0. a(2*n) = 0 unless n=0. a(2*n - 1) = A058483(n).

Example

G.f. = 1/q + 2 - q + 7*q^3 - 9*q^5 + 10*q^7 - 23*q^9 + 38*q^11 - 47*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1/q) (QPochhammer[ q^2]^3 QPochhammer[ q^6]^3 / (QPochhammer[ q] QPochhammer[ q^3] QPochhammer[ q^4]^2 QPochhammer[ q^12]^2))^2, {q, 0, n}]; (* Michael Somos, Sep 05 2014 *)
a[ n_] := SeriesCoefficient[ (1/q) (QPochhammer[ -q] QPochhammer[ -q^3] / (QPochhammer[ q^4] QPochhammer[ q^12]))^2, {q, 0, n}]; (* Michael Somos, Sep 05 2014 *)
a[ n_] := SeriesCoefficient[ 4 EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3] / (EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3]), {q, 0, n}]; (* Michael Somos, Sep 05 2014 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ((eta(x^2 + A)^3 * eta(x^6 + A)^3) / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^2 * eta(x^12 + A)^2))^2, n))};

Crossrefs

Cf. A058483, A112165, A123861, A187196.

Sequence in context: A117651 A268728 A187196 * A174869 A330862 A269158

Adjacent sequences: A187194 A187195 A187196 * A187198 A187199 A187200

Keyword

sign

Author

Michael Somos, Mar 06 2011

Status

approved