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A107635
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McKay-Thompson series of class 32a for the Monster group.
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0
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1, 3, 3, 4, 9, 12, 15, 21, 30, 43, 54, 69, 94, 123, 153, 193, 252, 318, 391, 486, 609, 754, 918, 1119, 1376, 1680, 2019, 2432, 2946, 3540, 4220, 5034, 6015, 7157, 8463, 9999, 11835, 13956, 16374, 19206, 22542, 26376, 30750, 35829, 41745, 48526, 56250
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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
| D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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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
| Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^3)) where f(u, v)=u^4+v^4+8*u*v-u^3*v^3.
Euler transform of period 4 sequence [3, -3, 3, 0, ...].
G.f.: Product_{k>0} (1+(-x)^k)^-3.
Expansion of q^(1/8)(eta(q^2)^2/(eta(q)eta(q^4)))^3 in powers of q.
Expansion of chi(q)^3 in powers of q where chi() is a Ramanujan theta function.
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EXAMPLE
| T32a = 1/q +3*q^7 +3*q^15 +4*q^23 +9*q^31 +12*q^39 +15*q^47 +...
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)^2/eta(x+A)/eta(x^4+A))^3, n))}
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CROSSREFS
| Cf. A022598(n)=(-1)^n*a(n).
Sequence in context: A045794 A065678 A022598 * A132319 A130626 A175796
Adjacent sequences: A107632 A107633 A107634 * A107636 A107637 A107638
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, May 18 2005
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