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A093160
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Expansion of q^(-1/2) * (eta(q^4) / eta(q))^4 in powers of q.
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5
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1, 4, 14, 40, 101, 236, 518, 1080, 2162, 4180, 7840, 14328, 25591, 44776, 76918, 129952, 216240, 354864, 574958, 920600, 1457946, 2285452, 3548550, 5460592, 8332425, 12614088, 18953310, 28276968, 41904208, 61702876, 90304598
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OFFSET
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0,2
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
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A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 381, Section 488.
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LINKS
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Table of n, a(n) for n=0..30.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^4.
Expansion of q^(-1/2) * k / (4 * k') in powers of q where q is Jacobi's nome and k is the elliptic modulus.
Expansion of q^(-1/4) * k^(1/2) / (2 * (1 - k)) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.
Expansion of (psi(x^2) / phi(-x))^2 = (psi(x) / phi(-x^2))^4 = (psi(-x) / phi(-x))^4 = (psi(x^2) / psi(-x))^4 = (chi(x) / chi(-x^2)^2)^4 = ( chi(x) * chi(-x)^2)^-4 = (chi(-x) * chi(-x^2))^-4 = (f(-x^4) / f(-x))^4 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [4, 4, 4, 0, ...].
Given g.f. A(x), then B(x)= x * A(x^2) satisfies 0 = f(B(x), B(x^2)) where f(u, v)= u^2 - v - 16*u*v - 16*v^2 - 256*u*v^2.
G.f. A(q) satisfies A(q) = sqrt(A(-q^2)) / (1 - 4*q*A(-q^2)); together with limit_{n->infinity} A(x^n) = 1 this gives a fast algorithm to compute the series. [Joerg Arndt, Aug 06 2011]
A001938(n) = (-1)^n * a(n). Convolution inverse of A112143.
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EXAMPLE
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1 + 4*x + 14*x^2 + 40*x^3 + 101*x^4 + 236*x^5 + 518*x^6 + 1080*x^7 + 2162*x^8 + ...
q + 4*q^3 + 14*q^5 + 40*q^7 + 101*q^9 + 236*q^11 + 518*q^13 + 1080*q^15 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (Product[ 1 + q^k, {k, 2, n, 2}] / Product[ 1 - q^k, {k, 1, n, 2}])^4, {q, 0, n}]
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[m] / (4 Sqrt[1 - m]), {q, 0, n + 1/2}]]
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ m^(1/4) / (2 (1 - Sqrt @ m)), {q, 0, n/2 + 1/4 }]]
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PROG
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(PARI) {a(n) = local(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A2 = A * (1 + 16*A); A= 8*A2 + (1 + 32*A) * sqrt(A2)); polcoeff( sqrt(A/x), n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^4, n))}
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CROSSREFS
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Cf. A001938, A112143.
Sequence in context: A144141 A187594 A066375 * A001938 A066368 A160463
Adjacent sequences: A093157 A093158 A093159 * A093161 A093162 A093163
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KEYWORD
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nonn,easy
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AUTHOR
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Michael Somos, Mar 26 2004, Apr 17 2007
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STATUS
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approved
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