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%I #31 Mar 12 2021 22:24:47
%S 1,-4,4,0,4,-8,0,0,4,-4,8,0,0,-8,0,0,4,-8,4,0,8,0,0,-4,16,-28,8,-16,
%T 32,-8,0,-16,20,-32,8,0,36,-8,0,-16,40,-24,0,-32,0,-8,4,-16,64,-36,28,
%U -32,40,-8,16,-32,32,-32,8,-48,32,-8,16,-64,52,-16,32,0
%N Expansion of (phi(-q) * phi(-q^23))^2 in powers of q where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C a(n) differs from A104794(n) = (-1)^n*A004018(n) from a(23) = -4 on. - _M. F. Hasler_, Mar 08 2018
%H G. C. Greubel, <a href="/A253185/b253185.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 (eta(q) * eta(q^23))^4 / (eta(q^2) * eta(q^46))^2 in powers of q.
%F Euler transform of a period 46 sequence.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (46 t)) = 368 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253183.
%F G.f.: (Sum_{k in Z} q^k^2)^2 * (Sum_{k in Z} q^(23*k^2))^2.
%e G.f. = 1 - 4*q + 4*q^2 + 4*q^4 - 8*q^5 + 4*q^8 - 4*q^9 + 8*q^10 - 8*q^13 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^23])^4 / (QPochhammer[ q^2] QPochhammer[ q^46])^2, {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^23 + A))^4 / (eta(x^2 + A) * eta(x^46 + A))^2, n))};
%o (PARI) {A253185(n,o=O('x^(n+1)))= polcoeff(((eta('x+o)*eta('x^23+o))^2/(eta('x^2+o)*eta('x^46+o)))^2, n)} \\ Writing the g.f. as a square makes the code more than 2 x faster. Using 'x prevents erroneous results in case x is used elsewhere. - _M. F. Hasler_, Mar 08 2018
%o (PARI) A253185_vec(N)={my(q='q+O('q^N)); Vec((eta(q) * eta(q^23))^4 / (eta(q^2) * eta(q^46))^2)} \\ _Joerg Arndt_, Mar 09 2018
%Y Cf. A104794, A253183.
%K sign
%O 0,2
%A _Michael Somos_, Mar 24 2015
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