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A214263 Expansion of f(x^1, x^7) in powers of x where f() is Ramanujan's general theta function.

%I A214263 #39 Feb 16 2025 08:33:18
%S A214263 1,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,
%T A214263 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A214263 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0
%N A214263 Expansion of f(x^1, x^7) in powers of x where f() is Ramanujan's general theta function.
%C A214263 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A214263 Characteristic function of A074377: a(n) = 1 if and only if n is in A074377.
%H A214263 Seiichi Manyama, <a href="/A214263/b214263.txt">Table of n, a(n) for n = 0..10000</a>
%H A214263 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A214263 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%H A214263 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%F A214263 Euler transform of period 16 sequence [ 1, -1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, 1, -1, ...].
%F A214263 G.f.: f(x, x^7) = sum_{k in Z} x^(4*k^2 - 3*k).
%F A214263 a(n) = A010054(2*n + 1) = A115359(2*n).
%F A214263 Sum_{k=1..n} a(k) ~ sqrt(n). - _Amiram Eldar_, Jan 13 2024
%e A214263 G.f. = 1 + x + x^7 + x^10 + x^22 + x^27 + x^45 + x^52 + x^76 + x^85 + x^115 + ...
%e A214263 G.f. = q^9 + q^25 + q^121 + q^169 + q^361 + q^441 + q^729 + q^841 + q^1225 + ...
%t A214263 f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; Table[SeriesCoefficient[f[q, q^7], {q, 0, n}], {n, 0, 50}] (* _G. C. Greubel_, Oct 05 2017 *)
%o A214263 (PARI) {a(n) = issquare(16*n + 9)};
%Y A214263 Cf. A047621, A074377, A115359.
%Y A214263 A000122, A080995, A010054, A133100, A089801 have g.f. of f(x,x^k) for k=1..5.
%Y A214263 Cf. A000122, A000700, A121373.
%K A214263 nonn,easy
%O A214263 0,1
%A A214263 _Michael Somos_ and _Omar E. Pol_, Jul 09 2012