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 A107035 Expansion of q * (psi(q^4) / phi(-q))^2 in powers of q where phi(), psi() are Ramanujan theta functions. 10
 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; text; internal format)
 OFFSET 1,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). REFERENCES R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eqs. (20), (21), (24) LINKS Seiichi Manyama, Table of n, a(n) for n = 1..1000 Kevin Acres and David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10. Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions 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 omega). Expansion of elliptic (1/8) * (-1 + 1 / sqrt(1 - lambda(z)) = (1/8) * (-1 + 1 / k') in powers of the nome q = exp(Pi i z). Expansion of ((phi(q) / phi(-q))^2 - 1) / 8 in powers of q where phi() is a Ramanujan theta function. 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). 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. a(n) ~ exp(sqrt(2*n)*Pi) / (64 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015 EXAMPLE G.f. = q + 4*q^2 + 12*q^3 + 32*q^4 + 78*q^5 + 176*q^6 + 376*q^7 + 768*q^8 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ (1/4) (EllipticTheta[ 2, 0, q^2] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* Michael Somos, Jun 13 2012 *) a[ n_] := With[ {m = ModularLambda[ Log[q] / (Pi I)]}, SeriesCoefficient[ (1/8) (-1 + 1 / Sqrt[1 - m]), {q, 0, n}]]; (* Michael Somos, Jun 13 2012 *) nmax = 50; CoefficientList[Series[Product[(1 + x^k)^4 * (1 + x^(2*k))^2 * (1 + x^(4*k))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *) QP = QPochhammer; s = (QP[q^2]/QP[q^4])^2*(QP[q^8]/QP[q])^4 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *) PROG (PARI) {a(n) = my(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))}; CROSSREFS Cf. A014969, A131124, A131126. Sequence in context: A084566 A208903 A079769 * A260145 A260778 A118885 Adjacent sequences: A107032 A107033 A107034 * A107036 A107037 A107038 KEYWORD nonn AUTHOR Michael Somos, May 09 2005 STATUS approved

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Last modified August 7 14:24 EDT 2024. Contains 375013 sequences. (Running on oeis4.)