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A131125
McKay-Thompson series of class 8E for the Monster group with a(0) = 4.
3
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
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
KEYWORD
sign
AUTHOR
Michael Somos, Jun 15 2007
STATUS
approved