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 A138526 Expansion of phi(-q^5) / phi(-q) in powers of q where phi() is a Ramanujan theta function. 11
 1, 2, 4, 8, 14, 22, 36, 56, 84, 126, 184, 264, 376, 528, 732, 1008, 1374, 1856, 2492, 3320, 4394, 5784, 7568, 9848, 12756, 16442, 21096, 26960, 34312, 43500, 54956, 69184, 86804, 108576, 135392, 168336, 208722, 258096, 318320, 391632, 480664 (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). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of (eta(q^5) / eta(q))^2 * eta(q^2) / eta(q^10) in powers of q. Euler transform of period 10 sequence [ 2, 1, 2, 1, 0, 1, 2, 1, 2, 0, ...]. G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - v^2)^2 - u^2 * (v^2 - 1) * (5*v^2 - 1). G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u^2 - v^2) * (u + v)^2 - u * v * (u^2 - 1) * (5*v^2 - 1). G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 * w^2 - u * v * (v^2 - 1). G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 * u6 - u2 * u3)^2 - u1 * u3 * (u2^2 - u6^2). G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138532. G.f.: Product_{k>0} P(5, x^k) / P(10, x^k) where P(n, x) is the n-th cyclotomic polynomial. Convolution square is A138517. Convolution inverse is A138527. a(n) ~ exp(2*Pi*sqrt(n/5)) / (2*5^(3/4)*n^(3/4)). - Vaclav Kotesovec, Sep 01 2015 EXAMPLE G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 22*q^5 + 36*q^6 + 56*q^7 + 84*q^8 + ... MATHEMATICA nmax=50; CoefficientList[Series[Product[(1+x^k)*(1-x^(5*k))/((1-x^k)*(1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *) a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^5] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Sep 14 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^5 + A) / eta(x + A))^2 * eta(x^2 + A) / eta(x^10 + A), n))}; CROSSREFS Cf. A138517, A138527, A138532. Sequence in context: A231429 A259392 A261968 * A286522 A201347 A089054 Adjacent sequences:  A138523 A138524 A138525 * A138527 A138528 A138529 KEYWORD nonn AUTHOR Michael Somos, Mar 23 2008 STATUS approved

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Last modified September 24 06:13 EDT 2021. Contains 347623 sequences. (Running on oeis4.)