A131125 McKay-Thompson series of class 8E for the Monster group with a(0) = 4.

1, 4, 4, 0, 2, 0, -8, 0, -1, 0, 20, 0, -2, 0, -40, 0, 3, 0, 72, 0, 2, 0, -128, 0, -4, 0, 220, 0, -4, 0, -360, 0, 5, 0, 576, 0, 8, 0, -904, 0, -8, 0, 1384, 0, -10, 0, -2088, 0, 11, 0, 3108, 0, 12, 0, -4552, 0, -15, 0, 6592, 0, -18, 0, -9448, 0, 22, 0, 13392, 0, 26, 0, -18816, 0, -29, 0, 26216, 0, -34, 0

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 q^-1 * (phi(q) / psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of (eta(q^2)^5 / ( eta(q)^2* eta(q^4)* eta(q^8)^2 ))^2 in powers of q.

Euler transform of period 8 sequence [ 4, -6, 4, -4, 4, -6, 4, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (u-8) * (v-4) - (v-8)^2.

G.f.: (1/x)* Product_{k>0} (1 + x^k)^2 / ((1 + x^(2*k))^3 * (1 + x^(4*k))^2).

a(2*n - 1) = A029841(n). a(2*n) = 0 unless n = 0. a(4*n - 1) = A029839(n). a(4*n + 1) = 4 * A079006(n).

Example

G.f. = 1/q + 4 + 4*q +2*q^3 - 8*q^5 - q^7 + 20*q^9 - 2*q^11 - 40*q^13 + 3*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (2 EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(eta[q^2]^5/(eta[q]^2*eta[q^4]*eta[q^8]^2))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 17 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2))^2, n))};

Crossrefs

Cf. A029839, A029841, A079006.

Sequence in context: A283998 A318143 A131124 * A187149 A106508 A177036

Adjacent sequences: A131122 A131123 A131124 * A131126 A131127 A131128

Keyword

sign

Author

Michael Somos, Jun 15 2007

Status

approved