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Search: id:A131999
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| A131999 |
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Expansion of eta(q)^2 * eta(q^2) * eta(q^4)^3/ eta(q^8)^2 in powers of q. |
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1
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| 1, -2, -2, 4, -2, 8, 4, -16, -2, -14, 8, 20, 4, 24, -16, -16, -2, -36, -14, 36, 8, 32, 20, -48, 4, -42, 24, 40, -16, 56, -16, -64, -2, -40, -36, 64, -14, 72, 36, -48, 8, -84, 32, 84, 20, 56, -48, -96, 4, -114, -42, 72, 24, 104, 40, -80, -16, -72, 56, 116, -16, 120, -64, -112, -2, -96, -40, 132, -36, 96, 64, -144, -14, -148
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.67).
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FORMULA
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Expansion of phi(q) * phi(q^2) * phi(-q)^2 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 8 sequence [ -2, -3, -2, -6, -2, -3, -2, -4, ...].
G.f. A(x) satisfies 0= f(A(x), A(x^2), A(x^4)) where f(u, v, w)= v^4 +u^2*v^2 +2*u^2*w^2 +2*u*v*w* (-u+2*v-2*w) -2*u*v^3.
a(n)= 2*b(n) where b(n) is multiplicative with b(2^e) = 1, b(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 5 (mod 8).
a(2n)= a(n).
G.f.: 1 -2* Sum_{k>0} k* x^k/(1-x^k)* kronecker(2, k) = Product_{k>0} (1-x^k)^4* (1+x^k)^2* (1+x^(2k))/ (1+x^(4k))^2.
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PROGRAM
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(PARI) {a(n)= if(n<1, n==0, -2* sumdiv(n, d, d* kronecker(2, d)))}
(PARI) {a(n)= local(A, p, e); if(n<0, n==0, A=factor(n); -2* prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 1, if(abs(p%8-4)==3, (p^(e+1)-1)/(p-1), ((-p)^(e+1)-1)/(-p-1))))))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)^2* eta(x^2+A)* eta(x^4+A)^3/ eta(x^8+A)^2, n))}
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CROSSREFS
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-2*A117000(n)= a(n) unless n=0. A113416(n)= (-1)^n* a(n). -2*A113417(n)= a(2n+1).
Sequence in context: A096154 A084540 A113416 this_sequence A103178 A087909 A076078
Adjacent sequences: A131996 A131997 A131998 this_sequence A132000 A132001 A132002
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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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