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 A228864 Expansion of 1 + q * (psi(-q^5) / psi(-q))^2 in powers of q where psi() is a Ramanujan theta function. 2
 1, 1, 2, 3, 6, 11, 16, 24, 38, 57, 82, 117, 168, 238, 328, 448, 614, 834, 1114, 1480, 1966, 2592, 3384, 4398, 5704, 7361, 9436, 12045, 15344, 19470, 24576, 30922, 38822, 48576, 60548, 75259, 93342, 115454, 142360, 175104, 214958, 263262, 321584, 391993, 476952 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 (phi(q^5) / phi(q))^2 * (chi^5(q) / chi(q^5)) in powers of q where phi(), chi() are Ramanujan theta functions. Expansion of eta(q^10)^8 / (eta(q) * eta(q^4) * eta(q^5)^3 * eta(q^20)^3) in powers of q. Euler transform of period 20 sequence [ 1, 1, 1, 2, 4, 1, 1, 2, 1, -4, 1, 2, 1, 1, 4, 2, 1, 1, 1, 0, ...]. G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A225849. a(n) = A210458(n) unless n=0. a(n) = (-1)^n * A138520(n). a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017 EXAMPLE G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 11*x^5 + 16*x^6 + 24*x^7 + 38*x^8 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ 1 + (EllipticTheta[ 2, Pi/4, q^(5/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)])^2, {q, 0, n}]; (* Michael Somos, Oct 26 2015 *) a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^5] / EllipticTheta[ 3, 0, q])^2 QPochhammer[ q^5, -q^5] / QPochhammer[ q, -q]^5, {q, 0, n}]; (* Michael Somos, Oct 26 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^10 + A)^8 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A)^3 * eta(x^20 + A)^3), n))}; CROSSREFS Cf. A138520, A210458, A225849. Sequence in context: A138519 A138520 A210458 * A289434 A049794 A034031 Adjacent sequences: A228861 A228862 A228863 * A228865 A228866 A228867 KEYWORD nonn AUTHOR Michael Somos, Sep 05 2013 STATUS approved

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Last modified July 23 03:47 EDT 2024. Contains 374544 sequences. (Running on oeis4.)