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Search: id:A132000
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| A132000 |
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Expansion of eta(q) * eta(q^2)^4 * eta(q^3)^5/ eta(q^6)^4 in powers of q. |
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+0
2
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| 1, -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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0,3
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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 (1/3) * b(q) * b(q^2) * c(q)^2/ c(q^2) in powers of q where b(), c() are cubic AGM functions.
Expansion of phi(-q)^2* phi(-q^3)^2* psi(q)^3/ psi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 6 sequence [ -1, -5, -6, -5, -1, -6, ...].
a(n)= -b(n) where b(n) is multiplicative with b(2^e) = 2+((-4)^(e+1)-1)/5, b(3^e) = 1, b(p^e) = (q^(e+1)-1)/(q-1) where q = p^2*kronecker(-3,p) if p>3.
a(3n)= a(n).
G.f.: 1 - Sum_{k>0} k^2* kronecker(-3,k)* x^k/(1-(-x)^k) = 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, n==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( 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, n==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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A113261(n)= (-1)^n* a(n). Convolution of A123330 and A131943.
Sequence in context: A147414 A117637 A113261 this_sequence A132001 A063004 A146993
Adjacent sequences: A131997 A131998 A131999 this_sequence A132001 A132002 A132003
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Aug 06 2007
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