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 A093160 Expansion of q^(-1/2) * (eta(q^4) / eta(q))^4 in powers of q. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). REFERENCES A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 381, Section 488. LINKS Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA 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. EXAMPLE 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 + ... MATHEMATICA 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 }]] PROG (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))} CROSSREFS Cf. A001938, A112143. Sequence in context: A144141 A187594 A066375 * A001938 A066368 A160463 Adjacent sequences:  A093157 A093158 A093159 * A093161 A093162 A093163 KEYWORD nonn,easy AUTHOR Michael Somos, Mar 26 2004, Apr 17 2007 STATUS approved

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