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A193261
McKay-Thompson series of class 18D for the Monster group with a(0) = -2.
3
1, -2, 0, 1, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 0, -3, 0, 0, -1, 0, 0, 4, 0, 0, 4, 0, 0, 1, 0, 0, -4, 0, 0, -6, 0, 0, -1, 0, 0, 5, 0, 0, 8, 0, 0, 1, 0, 0, -8, 0, 0, -10, 0, 0, -2, 0, 0, 11, 0, 0, 14, 0, 0, 4, 0, 0, -14, 0, 0, -19, 0, 0, -4, 0, 0, 17, 0, 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 eta(q)^2 * eta(q^6) * eta(q^9) / (eta(q^2) * eta(q^3) * eta(q^18)^2) in powers of q.
Expansion of q^(-1) * phi(-q) / (chi(-q^3) * psi(q^9)) = -2 + q^(-1) * chi(-q^9)^3 / chi(-q^3) = -3 + q^(-1) * psi(q) / psi(q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of -2 + c(q^3) / c(q^6) in powers of q where c() is a cubic AGM theta function.
Euler transform of period 18 sequence [ -2, -1, -1, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -1, -1, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 6 * g(t) where q = exp(2 Pi i t) and g() is g.f. for A128129.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (u + 3) * (v + 2) - v * (v - u).
Convolution inverse of A128129.
a(3*n + 1) = 0. a(3*n) = 0 unless n=0. a(3*n - 1) = A062242(n). a(n) = A143840(n) unless n=0. a(6*n - 1) = A132179(n). a(6*n + 2) = A092848(n).
EXAMPLE
1/q - 2 + q^2 + q^5 - q^8 - q^11 + q^17 + 2*q^20 - 2*q^26 - 3*q^29 + ...
MATHEMATICA
QP = QPochhammer; s = QP[q]^2*QP[q^6]*QP[q^9]/(QP[q^2]*QP[q^3]*QP[q^18]^2) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) * eta(x^9 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^18 + A)^2), n))}
CROSSREFS
Essentially the same as A143840.
Sequence in context: A260942 A260162 A123858 * A283497 A265507 A035145
KEYWORD
sign
AUTHOR
Michael Somos, Jul 19 2011
STATUS
approved