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A132976
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McKay-Thompson series of class 36B for the Monster group with a(0) = -1.
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3
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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 eta(q) * eta(q^4) * eta(q^18) / (eta(q^2) * eta(q^9) * eta(q^36)) in powers of q.
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.
Euler transform of period 36 sequence [ -1, 0, -1, -1, -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, -1, -1, 0, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u3 * u6 - (u1 + u2 + u1*u2) * (u3 + u6 + 3).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 / f(t) where q = exp(2 pi i t).
a(3*n + 1) = 0. a(3*n) = 0 unless n=0. a(3*n - 1) = A062244(n). a(2*n) = -A139032(n). a(6*n - 1) = A132179(n). a(6*n + 2) = -A092848(n).
A143840(n) = -(-1)^n * a(n). Convolution inverse of A132975.
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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, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, 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 + A) * eta(x^4 + A) * eta(x^18 + A) / ( eta(x^2 + A) * eta(x^9 + A) * eta(x^36 + A) ), n))}
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CROSSREFS
| Cf. A062244, A092848, A132975, A139032, A143840.
Sequence in context: A045833 A117896 * A143840 A028649 A097798 A065205
Adjacent sequences: A132973 A132974 A132975 * A132977 A132978 A132979
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 07 2007
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