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Search: id:A132001
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| A132001 |
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Expansion of 1- (1/3)* b(q)* b(q^2)* c(q)^2/ c(q^2) in powers of q where b(), c() are cubic AGM functions. |
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| 1, 5, 1, -11, -24, 5, 50, 53, 1, -120, -120, -11, 170, 250, -24, -203, -288, 5, 362, 264, 50, -600, -528, 53, 601, 850, 1, -550, -840, -120, 962, 821, -120, -1440, -1200, -11, 1370, 1810, 170, -1272, -1680, 250, 1850, 1320, -24, -2640, -2208, -203, 2451, 3005, -288, -1870, -2808, 5, 2880
(list; graph; listen)
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OFFSET
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1,2
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.71).
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FORMULA
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Expansion of 1 -phi(-q)^2* phi(-q^3)^2* psi(q)^3/ psi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of 1 - eta(q)* eta(q^2)^4* eta(q^3)^5/ eta(q^6)^4 in powers of q.
a(n) is multiplicative with a(2^e) = 2+((-4)^(e+1)-1)/5, a(3^e) = 1, a(p^e) = (q^(e+1)-1)/(q-1) where q = p^2*kronecker(-3,p) if p>3.
a(3n)= a(n).
G.f.: Sum_{k>0} k^2* kronecker(-3,k)* x^k/(1-(-x)^k) = 1 - Product_{k>0} (1-x^(3k))* (1-x^k)^5/ (1-x^k+x^(2k))^4.
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PROGRAM
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(PARI) {a(n)= if(n<1, 0, -sumdiv(n, d, d^2* (-1)^d* kronecker(-3, d)))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( 1- eta(x+A)* eta(x^2+A)^4* eta(x^3+A)^5/ eta(x^6+A)^4, n))}
(PARI) {a(n)= local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3, 1, if(p==2, 2+((-4)^(e+1)-1)/5, p=p^2*kronecker(-3, p); (p^(e+1)-1)/(p-1))))))}
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CROSSREFS
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A132000(n)= -a(n) unless n=0.
Sequence in context: A117637 A113261 A132000 this_sequence A063004 A146993 A104572
Adjacent sequences: A131998 A131999 A132000 this_sequence A132002 A132003 A132004
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Aug 06 2007
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