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A105559
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McKay-Thompson series of class 6E for the Monster group with a(0) = 3.
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4
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1, 3, 6, 4, -3, -12, -8, 12, 30, 20, -30, -72, -46, 60, 156, 96, -117, -300, -188, 228, 552, 344, -420, -1008, -603, 732, 1770, 1048, -1245, -2976, -1776, 2088, 4908, 2900, -3420, -7992, -4658, 5460, 12756, 7408, -8583, -19944, -11564, 13344, 30756, 17744, -20448, -46944
(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).
A Hauptmodul for Gamma_0(6).
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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 (eta(q^2)eta(q^3)^3/(eta(q)eta(q^6)^3))^3 in powers of q.
G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)=v^2+8u+6uv-u^2v.
G.f.: x^-1(Product_{k>0} (1-x^(6k-3))^3/(1-x^(2k-1)))^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 8 g(t) where q = exp(2 pi i t) and g() is g.f. for A128643.
Expansion of (c(q)/c(q^2))^3 in powers of q where c() is a cubic AGM analog function.
Expansion of q^(-1)(chi(-q^3)^3/chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.
Euler transform of period 6 sequence [ 3, 0, -6, 0, 3, 0, ...].
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EXAMPLE
| 1/q + 3 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*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)^3/eta(x+A)/eta(x^6+A)^3 )^3, n))}
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CROSSREFS
| a(n)=A007258(n) unless n=0. Convolution inverse of A123633.
Sequence in context: A197071 A140072 A187148 * A090038 A006464 A159354
Adjacent sequences: A105556 A105557 A105558 * A105560 A105561 A105562
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Apr 13 2005, Jan 21 2009
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