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A107035
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Expansion of q * (psi(q^4) / phi(-q))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
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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; 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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REFERENCES
| R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eqs. (20),(21),(24)
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of (eta(q^2) / eta(q^4))^2 * (eta(q^8) / eta(q))^4 in powers of q.
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 the 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.
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 + 4 * v^2 + 8 * u * v + 32 * u * v^2.
G.f. x * Product_{k>0} (1 + x^k)^4 (1 + x^(2*k))^2 * (1 + x^(4*k))^4.
Convolution inverse of A131124. A131126(n) = 4 * a(n) unless n=0. A014969(n) = 8 * a(n) unless n=0.
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EXAMPLE
| 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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PROG
| (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
| Cf. A014969, A131124, A131126.
Sequence in context: A004403 A084566 A079769 * A118885 A097392 A090634
Adjacent sequences: A107032 A107033 A107034 * A107036 A107037 A107038
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, May 09 2005
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