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%I #10 Mar 12 2021 22:24:45
%S 1,-2,3,-5,7,-10,14,-20,27,-36,48,-63,82,-106,137,-175,222,-280,352,
%T -439,546,-676,834,-1024,1253,-1528,1857,-2250,2718,-3276,3936,-4718,
%U 5640,-6728,8006,-9507,11266,-13324,15726,-18526,21786,-25574,29970,-35064,40961
%N Expansion of q * chi(q^3) * chi(q^5) / (chi(q) * chi(q^15))^2 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 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 (eta(q) * eta(q^4) * eta(q^6) * eta(q^10) * eta(q^15) * eta(q^60))^2 / (eta(q^2)^4 * eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20) * eta(q^30)^4) in powers of q.
%F Euler transform of a period 60 sequence.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = f(t) where q = exp(2 Pi i t).
%F a(n) = - A145728(n) unless n=0. a(n) = -(-1)^n * A123630(n).
%F Convolution inverse of A145788.
%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(2*n/15)) / (2^(7/4) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017
%e G.f. = q - 2*q^2 + 3*q^3 - 5*q^4 + 7*q^5 - 10*q^6 + 14*q^7 - 20*q^8 + 27*q^9 + ...
%t a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^3, x^6] QPochhammer[ -x^5, x^10] / (QPochhammer[ -x, x^2] QPochhammer[ -x^15, x^30])^2 , {x, 0, n}]; (* _Michael Somos_, Sep 03 2015 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) * eta(x^60 + A))^2 / (eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A) * eta(x^30 + A)^4), n))};
%Y Cf. A123630, A145728, A145788.
%K sign
%O 1,2
%A _Michael Somos_, Oct 23 2008
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