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 A224916 Expansion of chi(x)^2 / chi(-x^2)^6 in powers of x where chi() is a Ramanujan theta function. 1
 1, 2, 7, 14, 31, 58, 112, 196, 347, 580, 966, 1554, 2485, 3872, 5993, 9102, 13719, 20384, 30068, 43836, 63481, 91048, 129763, 183448, 257839, 359862, 499583, 689312, 946416, 1292388, 1756838, 2376598, 3201557, 4293942, 5736736, 7633702, 10121408, 13370634 (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 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of q^(-5/12) * (eta(q^4)^2 / (eta(q) * eta(q^2)))^2 in powers of q. Expansion of psi(x^2)^2 / f(-x)^2 = 1 / (chi(-x)^2 * chi(-x^2)^4) = 1 / (chi(x)^4 * chi(-x)^6 ) in powers of x where psi(), chi(), f() are Ramanujan theta functions. Expansion of (chi(x)^4 - chi(-x)^4) / (8*x) in powers of x^2 where chi() is a Ramanujan theta function. Euler transform of period 4 sequence [ 2, 4, 2, 0, ...]. G.f.: Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k))^4. G.f.: (Sum_{k>0} x^(k^2 - k)) / (Product_{k>0} (1 - x^k))^2. - Michael Somos, Jul 04 2013 a(n) = A112160(2*n + 1) / 4. Convolution square of A098613. - Michael Somos, Jul 04 2013 a(n) ~ exp(2*Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015 EXAMPLE 1 + 2*x + 7*x^2 + 14*x^3 + 31*x^4 + 58*x^5 + 112*x^6 + 196*x^7 + 347*x^8 + ... q^5 + 2*q^17 + 7*q^29 + 14*q^41 + 31*q^53 + 58*q^65 + 112*q^77 + 196*q^89 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^2 / (4 q^(1/2) QPochhammer[q]^2), {q, 0, n}] a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ q^2, q^4]^4 / QPochhammer[ q, q^2]^2, {q, 0, n}] a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2]^4 - QPochhammer[ q, q^2]^4)/ 8, {q, 0, 2 n + 1}] PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^2 / (eta(x + A) * eta(x^2 + A)))^2, n))} CROSSREFS Cf. A098613, A112160. Sequence in context: A221319 A221320 A221235 * A258321 A034791 A140253 Adjacent sequences:  A224913 A224914 A224915 * A224917 A224918 A224919 KEYWORD nonn AUTHOR Michael Somos, Apr 19 2013 STATUS approved

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Last modified February 26 05:36 EST 2020. Contains 332276 sequences. (Running on oeis4.)