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A134746
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Expansion of 1+k in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.
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0
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1, 4, 0, -16, 0, 56, 0, -160, 0, 404, 0, -944, 0, 2072, 0, -4320, 0, 8648, 0, -16720, 0, 31360, 0, -57312, 0, 102364, 0, -179104, 0, 307672, 0, -519808, 0, 864960, 0, -1419456, 0, 2299832, 0, -3682400, 0, 5831784, 0, -9141808, 0, 14194200, 0, -21842368, 0, 33329700
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of (phi(q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.
Expansion of (eta(q^8) / eta(q))^4 * (eta(q^2) / eta(q^4))^14 in powers of q.
Euler transform of period 8 sequence [ 4, -10, 4, 4, 4, -10, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 / f(t) where q = exp(2 pi i t).
a(2*n) = 0 unless n=0.
G.f.: ( (Sum_{k} x^(k^2)) / (Sum_{k} x^(2*k^2)) )^2 = ( Product_{k>0} (1 + x^k)^2 * (1 + x^(4*k))^2 / (1 + x^(2*k))^5 )^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v)^2 - u * (2 - u) * v^2.
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EXAMPLE
| 1 + 4*q - 16*q^3 + 56*q^5 - 160*q^7 + 404*q^9 - 944*q^11 + 2072*q^13 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( (eta(x^8 + A) / eta(x + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^7 )^2, n))}
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CROSSREFS
| 4 * A001938(n) = A127393(n) = a(2*n+1).
Sequence in context: A170771 A170772 A079986 * A199572 A003195 A190759
Adjacent sequences: A134743 A134744 A134745 * A134747 A134748 A134749
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Nov 07 2007
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