A145740 McKay-Thompson series of class 20C for the Monster group with a(0) = -2.

1, -2, 1, -2, 2, 2, -1, 0, -4, 2, 5, -2, 0, -8, 2, 8, -3, 2, -14, 6, 14, -6, 4, -24, 12, 24, -11, 4, -40, 16, 38, -16, 5, -62, 24, 60, -24, 10, -94, 40, 91, -38, 18, -144, 62, 136, -57, 24, -214, 88, 201, -82, 30, -308, 122, 288, -117, 48, -440, 180, 410, -168

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q) * eta(q^4) * eta(q^10) / (eta(q^2) * eta(q^5) * eta(q^20)))^2 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 5 / f(t) where q = exp(2 Pi i t).

Expansion of q^(-1) * (psi(-q) / psi(-q^5))^2 in powers of q where psi() is a Ramanujan theta function.

a(n) = A112159(n) = A225849(n) unless n=0.

a(2*n) = -2 * A138522(n).

Convolution square of A145708.

a(n) = -(-1)^n * A138516(n). - Michael Somos, Sep 04 2015

Example

G.f. = 1/q - 2 + q - 2*q^2 + 2*q^3 + 2*q^4 - q^5 - 4*q^7 + 2*q^8 + 5*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(5/2)])^2, {q, 0, n}]; (* Michael Somos, Sep 04 2015 *)
QP = QPochhammer; s = (QP[q]*QP[q^4]*(QP[q^10]/(QP[q^2]*QP[q^5]*QP[q^20]) ))^2 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)))^2, n))};

Crossrefs

Cf. A112159, A138561, A138522, A145708, A225849.

Sequence in context: A337474 A104605 A300953 * A138516 A180580 A026513

Adjacent sequences: A145737 A145738 A145739 * A145741 A145742 A145743

Keyword

sign

Author

Michael Somos, Oct 17 2008

Status

approved