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A132130
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McKay-Thompson series of class 10D for the Monster group with a(0) = 6.
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1
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1, 6, 21, 62, 162, 378, 819, 1680, 3276, 6138, 11145, 19662, 33840, 57048, 94362, 153432, 245757, 388218, 605466, 933414, 1423614, 2149586, 3215844, 4769544, 7016572, 10243896, 14848809, 21378276, 30582360, 43484304, 61473438, 86428896
(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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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
| Euler transform of period 10 sequence [ 6, 0, 6, 0, 0, 0, 6, 0, 6, 0, ...].
G.f. A(x) satisfies 0= f(A(x), A(x^2), A(x^4)) where f(u, v, w)= (v -u^2)*(v -w^2) -u*w* (12*(1+v^2) -20*v).
G.f. is Fourier series of a weight 0 level 10 modular form. f(-1/ ( 10 t)) = f(t) where q = exp(2 pi i t).
G.f.: x^(-1)* (Product_{k>0} (1+x^k)/ (1+x^(5*k)))^6.
G.f.: 1 / ( x * Product_{k>0} P(10,x^k)^6 ) where P(n,x) is the n-th cyclotomic polynomial.
Expansion of q^(-1) * (chi(-q^5) / chi(-q))^6 in powers of q where chi() is a Ramanujan theta function.
Expansion of (eta(q^2) * eta(q^5) / (eta(q) * eta(q^10)))^6 in powers of q.
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EXAMPLE
| 1/q + 6 + 21*q + 62*q^2 + 162*q^3 + 378*q^4 + 819*q^5 + 1680*q^6 + ...
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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^5+A)/ eta(x+A)/ eta(x^10+A))^6, n))}
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CROSSREFS
| A058100(n)= a(n) unless n=0.
Sequence in context: A012593 A048476 A122678 * A022571 A117962 A105457
Adjacent sequences: A132127 A132128 A132129 * A132131 A132132 A132133
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Aug 11 2007, Aug 09 2008
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), May 16 2008 at the suggestion of R. J. Mathar
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