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 A261252 Expansion of f(-x^3) * f(-x^6) / (f(x) * f(-x^4)) in powers of x where f() is a Ramanujan theta function. 2
 1, -1, 2, -4, 7, -10, 14, -22, 33, -45, 62, -88, 122, -163, 216, -290, 386, -502, 650, -846, 1093, -1393, 1768, -2248, 2844, -3565, 4454, -5566, 6927, -8566, 10562, -13014, 15986, -19543, 23832, -29032, 35272, -42700, 51578, -62226, 74906, -89909, 107712 (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..2500 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of f(-x^3)^3 / (f(-x^2)^2 * f(x, x^2)) in powers of x where f(,) is a Ramanujan theta function. Expansion of (chi(-x^3)^3 / chi(-x)) * (psi(x^3) / psi(x))^2 in powers of x where chi(), psi() are Ramanujan theta functions. Expansion of q^(-1/6) * eta(q) * eta(q^3) * eta(q^6) / eta(q^2)^3 in powers of q. Euler transform of period 6 sequence [ -1, 2, -2, 2, -1, 0, ...]. Given g.f. A(x), then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (3*u^2 - v)^3 * v^3 - 4 * u^2 * (v - u^2) * (2*u^2 - v). a(n) = A261251(3*n). Convolution inverse is A132179. a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (6^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018 EXAMPLE G.f. = 1 - x + 2*x^2 - 4*x^3 + 7*x^4 - 10*x^5 + 14*x^6 - 22*x^7 + ... G.f. = q - q^7 + 2*q^13 - 4*q^19 + 7*q^25 - 10*q^31 + 14*q^37 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^6] / (QPochhammer[ -x] QPochhammer[ x^4]), {x, 0, n}]; PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x^2 + A)^3, n))}; CROSSREFS Cf. A132179, A261251. Sequence in context: A288243 A170890 A079963 * A056750 A171926 A171930 Adjacent sequences:  A261249 A261250 A261251 * A261253 A261254 A261255 KEYWORD sign AUTHOR Michael Somos, Aug 12 2015 STATUS approved

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Last modified September 19 19:41 EDT 2021. Contains 347564 sequences. (Running on oeis4.)