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 A229180 Expansion of (chi(-x) * chi(-x^3))^-3 in powers of x where chi() is a Ramanujan theta function. 3
 1, 3, 6, 16, 33, 60, 118, 210, 354, 612, 1008, 1608, 2583, 4035, 6174, 9448, 14196, 21024, 31054, 45282, 65322, 93884, 133638, 188640, 265225, 370086, 512934, 708136, 971628, 1325724, 1802134, 2437200, 3280452, 4400132, 5876184, 7815288, 10360890, 13683525 (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). In Verrill (1999) section 2.6, denoted by g as a function of q. REFERENCES H. Verrill, Some Congruences related to modular forms, Max Planck Institute, 1999. LINKS G. C. Greubel, Table of n, a(n) for n = 0..2500 Michael Somos, Introduction to Ramanujan theta functions H. Verrill, Some Congruences related to modular forms Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of q^(-1/2) * (eta(q^2) * eta(q^6) / (eta(q) * eta(q^3)))^3 in powers of q. Euler transform of period 6 sequence [3, 0, 6, 0, 3, 0, ...]. G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = (1/8) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058492. G.f.: t / (1 - 10*t^2 + 9*t^4)^(1/2) where t = the g.f. of A217786. G.f.: 1 / (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(6*k - 3)))^3. Convolution inverse of A058492. a(n) ~ exp(2*Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015 EXAMPLE G.f. = 1 + 3*x + 6*x^2 + 16*x^3 + 33*x^4 + 60*x^5 + 118*x^6 + 210*x^7 + ... G.f. = q + 3*q^3 + 6*q^5 + 16*q^7 + 33*q^9 + 60*q^11 + 118*q^13 + 210*q^15 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[x^3, x^6])^3, {x, 0, n}]; nmax = 40; CoefficientList[Series[Product[1/((1 - x^(2*k - 1)) * (1 - x^(6*k - 3)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A) / (eta(x + A) * eta(x^3 + A)))^3, n))}; CROSSREFS Cf. A058492, A084471, A217786. Sequence in context: A291611 A308401 A196261 * A219810 A122742 A052370 Adjacent sequences: A229177 A229178 A229179 * A229181 A229182 A229183 KEYWORD nonn AUTHOR Michael Somos, Sep 30 2013 STATUS approved

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Last modified March 24 08:04 EDT 2023. Contains 361455 sequences. (Running on oeis4.)