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A223903
McKay-Thompson series of class 20C for the Monster group with a(0) = -1.
3
1, -1, 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
OFFSET
-1,4
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) * chi(q^5)^5 / chi(q) in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q) * eta(q^4) * eta(q^10)^10 / (eta(q^2)^2 * eta(q^5)^5 * eta(q^20)^5) in powers of q.
Euler transform of period 20 sequence [ -1, 1, -1, 0, 4, 1, -1, 0, -1, -4, -1, 0, -1, 1, 4, 0, -1, 1, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 1) * (u + 4) * v * (v - 1) * (v + 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A225701. - Michael Somos, Sep 04 2013
G.f.: (1/x) * Product_{k>0} (1 + x^(10*k - 5))^5 / (1 + x^(2*k - 1)).
a(n) = A112159(n) = A145740(n) unless n=0. a(n) = -(-1)^n * A132980(n).
EXAMPLE
G.f. = 1/q - 1 + 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[ (1/q) QPochhammer[ -q^5, q^10]^5 / QPochhammer[ -q, q^2], {q, 0, n}];
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)^10 / (eta(x^2 + A)^2 * eta(x^5 + A)^5 * eta(x^20 + A)^5), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 29 2013
STATUS
approved