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A132321
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McKay-Thompson series of class 30C for the Monster group with a(0) = -1.
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2
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1, -1, 0, -2, 2, -2, 3, -2, 5, -6, 5, -6, 9, -10, 10, -16, 17, -18, 25, -26, 31, -38, 37, -48, 60, -62, 68, -84, 95, -104, 125, -134, 154, -182, 192, -220, 257, -274, 309, -360, 394, -434, 492, -544, 607, -688, 740, -824, 944, -1018, 1123, -1266, 1377, -1524, 1697, -1850, 2041, -2264, 2461, -2708
(list; graph; refs; listen; history; internal format)
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OFFSET
| -1,4
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (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^3) * eta(q^5) * eta(q^15) / (eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30)) in powers of q.
Euler transform of period 30 sequence [ -1, 0, -2, 0, -2, 0, -1, 0, -2, 0, -1, 0, -1, 0, -4, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -2, 0, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 4 * u + 2 * u * v.
G.f. is a Fourier series which satisfies f(-1 / (30 t)) = 4 / f(t) where q = exp(2 pi i t).
G.f.: x^-1 * (Product_{k>0} (1+x^k) * (1+x^(3*k)) * (1+x^(5*k)) * (1+x^(15*k)))^-1.
Expansion of q^-1 * chi(-q) * chi(-q^3) * chi(-q^5) * chi(-q^15) in powers of q where chi() is a Ramanujan theta function.
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EXAMPLE
| q^-1 - 1 - 2*q^2 + 2*q^3 - 2*q^4 + 3*q^5 - 2*q^6 + 5*q^7 - 6*q^8 + ...
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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^3+A) * eta(x^5+A) * eta(x^15+A) / eta(x^2+A) / eta(x^6+A) / eta(x^10+A) / eta(x^30+A), n))}
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CROSSREFS
| A058614(n) = a(n) unless n = 0.
A058614(n) = a(n) unless n = 0. Convolution inverse of A123632.
Sequence in context: A104239 A058614 A058726 * A122765 A131053 A125600
Adjacent sequences: A132318 A132319 A132320 * A132322 A132323 A132324
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Aug 18 2007
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