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A143840
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McKay-Thompson series of class 18D for the Monster group with a(0) = 1.
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2
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1, 1, 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, 24, 0, 0, 4, 0, 0, -23
(list; graph; refs; listen; history; internal format)
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OFFSET
| -1,22
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of psi(q) / (q * psi(q^9)) = 1 + chi(-q^9)^3 / (q * chi(-q^3)) in powers of q where psi(), chi() are Ramanujan theta functions.
Expansion of eta(q^2)^2 * eta(q^9) / (eta(q) * eta(q^18)^2) in powers of q.
Euler transform of period 18 sequence [ 1, -1, 1, -1, 1, -1, 1, -1, 0, -1, 1, -1, 1, -1, 1, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1/ (18 t)) = 3 g(t) where q = exp(2 pi i t) and g() is g.f. for A128770.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (u^2 + 3) - (u + v)^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (u^2 - 3*u + 3) * (v^2 - 3*v + 3) - v^3.
a(3*n + 1) = 0. a(3*n) = 0 unless n=0. a(3*n - 1) = A062242(n). a(2*n) = A139032(n). a(6*n + 2) = A092848(n). a(6*n - 1) = A132179(n).
A132976(n) = -(-1)^n * a(n). Convolution inverse of A124243.
G.f.: 1 + x^(-1) * Product_{k>0} (1 - x^(18*k - 9))^3 / (1 - x^(6*k - 3)).
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EXAMPLE
| 1/q + 1 + q^2 + q^5 - q^8 - q^11 + q^17 + 2*q^20 - 2*q^26 - 3*q^29 + ...
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MATHEMATICA
| a[ n_] := If[ n < -1, 0, SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(9/2)], {q, 0, n}]]
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PROG
| (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^9 + A) / (eta(x + A) * eta(x^18 + A)^2), n))}
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CROSSREFS
| Cf. A062242, A092848, A128770, A124243, A132179, A132976, A139032.
Sequence in context: A045833 A117896 A132976 * A028649 A097798 A065205
Adjacent sequences: A143837 A143838 A143839 * A143841 A143842 A143843
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 02 2008
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