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 A227401 Expansion of chi(x^6) / (chi(-x) * chi(x^3)) in powers of x which chi() is a Ramanujan theta function. 1
 1, 1, 1, 1, 1, 2, 4, 5, 5, 5, 6, 9, 12, 15, 16, 17, 20, 26, 34, 40, 44, 48, 55, 68, 84, 98, 108, 118, 135, 161, 192, 221, 244, 268, 303, 354, 414, 470, 519, 571, 641, 737, 847, 954, 1052, 1156, 1291, 1465, 1664, 1861, 2048, 2248, 2496, 2807, 3158, 3511, 3855 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 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 Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016. Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of f(x^6) / f(-x^1, -x^5) in powers of x where f(,) is a Ramanujan theta function. Expansion of q^(1/12) * eta(q^2) * eta(q^3) * eta(q^12)^3 / (eta(q) * eta(q^6)^3 * eta(q^24)) in powers of q. Euler transform of period 24 sequence [ 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, -1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 0, ...]. a(n) ~ 11^(1/4) * exp(Pi*sqrt(11*n)/6) / (4*sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Jul 11 2016 EXAMPLE G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 5*x^7 + 5*x^8 + 5*x^9 + ... G.f. = 1/q + q^11 + q^23 + q^35 + q^47 + 2*q^59 + 4*q^71 + 5*q^83 + 5*q^95 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ -x^6] / (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6]), {x, 0, n}]; a[ n_] := SeriesCoefficient[ QPochhammer[ -x^6, x^12] QPochhammer[ -x, x] QPochhammer[ x^3, -x^3], {x, 0, n}]; PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^3 / (eta(x + A) * eta(x^6 + A)^3 * eta(x^24 + A)), n))}; CROSSREFS Sequence in context: A034214 A317749 A253415 * A131813 A083038 A061008 Adjacent sequences:  A227398 A227399 A227400 * A227402 A227403 A227404 KEYWORD nonn AUTHOR Michael Somos, Sep 20 2013 STATUS approved

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Last modified May 13 11:22 EDT 2021. Contains 343839 sequences. (Running on oeis4.)