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A138688
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McKay-Thompson series of class 24I for the Monster group with a(0) = 2.
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0
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1, 2, 4, 6, 11, 18, 28, 42, 62, 90, 128, 180, 250, 342, 464, 624, 831, 1098, 1440, 1878, 2432, 3132, 4012, 5112, 6485, 8190, 10300, 12900, 16097, 20016, 24804, 30636, 37724, 46314, 56700, 69228, 84302, 102402, 124088, 150024, 180973, 217836
(list; graph; refs; listen; history; internal format)
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OFFSET
| -1,2
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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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REFERENCES
| J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339. See page 335.
K. Bringmann and H. Swisher, On a conjecture of Koike on identities between Thompson series and Roger-Ramanujan functions, Proc. Amer. Math. Soc. 135 (2007), 2317-2326. See page 2325 (A.7)
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group
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FORMULA
| Expansion of psi(q^4) * phi(-q^3) / (phi(-q) * psi(q^12)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^3)^2 * eta(q^8)^2 * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6) * eta(q^24)^2) in powers of q.
Euler transform of period 24 sequence [ 2, 1, 0, 2, 2, 0, 2, 0, 0, 1, 2, 0, 2, 1, 0, 0, 2, 0, 2, 2, 0, 1, 2, 0, ...].
G.f.: (G(x) * G(x^24) + x^5 * H(x) * H(x^24))^2 * (G(x^4) * G(x^6) + x^2 * H(x^4) * H(x^6)) where G() is g.f. of A003114 and H() is g.f. of A003106.
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 pi i t).
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EXAMPLE
| 1/q + 2 + 4*q + 6*q^2 + 11*q^3 + 18*q^4 + 28*q^5 + 42*q^6 + 62*q^7 + ...
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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) * eta(x^3 + A)^2 * eta(x^8 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)^2), n))}
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CROSSREFS
| A058579(n) = a(n) unless n=0.
Sequence in context: A034297 A026636 A026658 * A131298 A168445 A185192
Adjacent sequences: A138685 A138686 A138687 * A138689 A138690 A138691
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Mar 26 2008
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