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%I #35 Mar 12 2021 22:24:46
%S 1,-3,9,-22,48,-99,194,-363,657,-1155,1977,-3312,5443,-8787,13968,
%T -21894,33873,-51795,78345,-117312,174033,-255945,373353,-540486,
%U 776848,-1109040,1573209,-2218198,3109713,-4335840,6014123,-8300811,11402928
%N Expansion of (psi(x^2) / psi(x))^3 in powers of x where psi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
%H Seiichi Manyama, <a href="/A187053/b187053.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2500 from G. C. Greubel)
%H A. Cayley, <a href="/A001934/a001934.pdf">A memoir on the transformation of elliptic functions</a>, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-3/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^3 in powers of q.
%F Euler transform of period 4 sequence [-3, 6, -3, 0, ...].
%F G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k-1)))^3.
%F Convolution inverse of A029840. Convolution cube of A083365. a(n) = (-1)^n * A001937(n).
%F a(n) ~ (-1)^n * 3^(1/4) * exp(sqrt(3*n/2)*Pi) / (16*2^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017
%e G.f. = 1 - 3*x + 9*x^2 - 22*x^3 + 48*x^4 - 99*x^5 + 194*x^6 - 363*x^7 + ...
%e G.f. = q^3 - 3*q^11 + 9*q^19 - 22*q^27 + 48*q^35 - 99*q^43 + 194*q^51 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ -x])^3, {x, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^3, n))};
%Y Cf. A001937, A029840, A083365.
%K sign
%O 0,2
%A _Michael Somos_, Mar 06 2011
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