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%I #14 Mar 12 2021 22:24:45
%S 1,1,0,1,1,0,0,1,1,1,1,1,2,1,1,1,2,2,1,2,3,3,2,3,4,3,2,4,5,4,4,5,6,6,
%T 5,6,8,7,6,8,11,10,8,11,13,11,10,13,16,15,14,17,20,18,17,20,24,23,21,
%U 25,31,29,26,32,37,34,32,39,44,42,41,47,54,52,49
%N Expansion of chi(q) * chi(-q^5) in powers of q where chi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A139632/b139632.txt">Table of n, a(n) for n = 0..1000</a>
%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^(1/4) * eta(q^2)^2 * eta(q^5) / (eta(q) * eta(q^4) * eta(q^10)) in powers of q.
%F Euler transform of period 20 sequence [ 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 1, 0, 1, -1, 0, 0, 1, -1, 1, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139631.
%F G.f.: Product_{k>0} (1 + x^k) / ((1 + x^(2*k)) * (1 + x^(5*k))).
%F a(n) = (-1)^floor((n + 1)/2) * A145705(n). - _Michael Somos_, Sep 07 2015
%F a(2*n) = A139631(n). a(2*n + 1) = A145703(n). - _Michael Somos_, Sep 07 2015
%F a(n) ~ exp(Pi*sqrt(n/10)) / (2^(5/4) * 5^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015
%e G.f. = 1 + x + x^3 + x^4 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + x^13 + ...
%e G.f. = 1/q + q^3 + q^11 + q^15 + q^27 + q^31 + q^35 + q^39 + q^43 + 2*q^47 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^5, x^10], {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t nmax = 100; CoefficientList[Series[Product[(1 + x^k) / ((1 + x^(2*k)) * (1 + x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^5 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)), n))};
%Y Cf. A139631, A145703, A145705.
%K nonn
%O 0,13
%A _Michael Somos_, Apr 27 2008
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