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Search: id:A107035
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| A107035 |
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Expansion of (eta(q^2)/eta(q^4))^2(eta(q^8)/eta(q))^4 in powers of q. |
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+0
4
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| 1, 4, 12, 32, 78, 176, 376, 768, 1509, 2872, 5316, 9600, 16966, 29408, 50088, 83968, 138738, 226196, 364284, 580032, 913824, 1425552, 2203368, 3376128, 5130999, 7738136, 11585208, 17225472, 25444278, 37350816, 54504160, 79085568
(list; graph; listen)
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OFFSET
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1,2
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REFERENCES
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R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eqs. (20),(21),(24)
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FORMULA
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Euler transform of period 8 sequence [4, 2, 4, 4, 4, 2, 4, 0, ...].
G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)=v-u^2+4v^2+8uv+32uv^2.
G.f. x(Product_{k>0} (1+x^k)^4 (1+x^(2k))^2 (1+x^(4k))^4).
Expansion of Fricke tau_8(omega)/16 in powers of q = exp(2*pi*i*z).
Expansion of elliptic -1+1/(8*sqrt(1-lambda(z)))=-1+1/(8*k') in powers of nome q = exp(pi*i*z).
Elliptic j(z) = 256*(x^4+8*x^3+20*x^2+16*x+1)^3/(x*(x+4)*(x+2)^2) where x= tau_8(z).
Expansion of ((phi(q) / phi(-q))^2 - 1) / 8 in powers of q where phi() is a Ramanujan theta function.
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EXAMPLE
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q + 4*q^2 + 12*q^3 + 32*q^4 + 78*q^5 + 176*q^6 + 376*q^7 + 768*q^8 + ...
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PROGRAM
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(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x^4+A))^2*(eta(x^8+A)/eta(x+A))^4, n))}
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CROSSREFS
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A131126(n) = 4 * a(n) unless n=0. A014969(n) = 8 * a(n) unless n=0.
Sequence in context: A004403 A084566 A079769 this_sequence A118885 A097392 A090634
Adjacent sequences: A107032 A107033 A107034 this_sequence A107036 A107037 A107038
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, May 09 2005
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