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%I #17 Jan 13 2024 03:32:51
%S 1,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,
%T 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
%U 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Expansion of f(x^3, x^5) in powers of x where f() is Ramanujan's two-variable 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="/A214264/b214264.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>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Euler transform of period 16 sequence [ 0, 0, 1, 0, 1, -1, 0, -1, 0, -1, 1, 0, 1, 0, 0, -1, ...].
%F G.f.: Sum_{k} x^(((8*k + 1)^2 - 1) / 16).
%F Characteristic function of A074378. a(n) = 1 if and only if n is in A074378.
%F a(n) = A010054(2*n).
%F Sum_{k=1..n} a(k) ~ sqrt(n). - _Amiram Eldar_, Jan 13 2024
%e 1 + x^3 + x^5 + x^14 + x^18 + x^33 + x^39 + x^60 + x^68 + x^95 + x^105 +
%e q + q^49 + q^81 + q^225 + q^289 + q^529 + q^625 + q^961 + q^1089 + ...
%t f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A214264[n_] := SeriesCoefficient[f[x^3, x^5], {x, 0, n}]; Table[A214264[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 03 2017 *)
%o (PARI) {a(n) = issquare( 16*n + 1)}
%Y Cf. A010054, A047522, A074378.
%Y Cf. A000122, A000700, A121373.
%K nonn
%O 0,1
%A _Michael Somos_, Jul 09 2012
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