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A205633 Expansion of f(x^3, x^7) in powers of x where f() is Ramanujan's two-variable theta function.

%I A205633 #42 Aug 30 2025 16:55:50
%S A205633 1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
%T A205633 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A205633 0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0
%N A205633 Expansion of f(x^3, x^7) in powers of x where f() is Ramanujan's two-variable theta function.
%C A205633 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A205633 G. C. Greubel, <a href="/A205633/b205633.txt">Table of n, a(n) for n = 0..1000</a>
%H A205633 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A205633 S. Cooper and M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00079-7">Results of Hurwitz type for three squares.</a> Discrete Math. 274 (2004), no. 1-3, 9-24. See D(q).
%H A205633 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A205633 Euler transform of period 20 sequence [ 0, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, 0, -1, ...]. - _Michael Somos_, Sep 22 2012
%F A205633 Characteristic function of A036666: a(n) = 1 if and only if n is in A036666.
%F A205633 G.f.: Sum_{k in Z} x^(5*k^2 + 2*k).
%F A205633 a(3*n + 2) = a(4*n + 1) = a(4*n + 2) = 0. a(4*n + 3) = A205988(n). - _Michael Somos_, Sep 22 2012
%e A205633 G.f. = 1 + x^3 + x^7 + x^16 + x^24 + x^39 + x^51 + x^72 + x^88 + x^115 + x^135 + ...
%e A205633 G.f. = q + q^16 + q^36 + q^81 + q^121 + q^196 + q^256 + q^361 + q^441 + q^576 + ...
%t A205633 A205633[n_] := SeriesCoefficient[QPochhammer[-q^3, q^10]* QPochhammer[-q^7, q^10]*QPochhammer[q^10, q^10], {q, 0, n}]; Table[A205633[n], {n,0,50}] (* _G. C. Greubel_, Jun 16 2017 *)
%o A205633 (PARI) {a(n) = issquare(5*n + 1)}; /* _Michael Somos_, Sep 22 2012 */
%Y A205633 Cf. A036666, A205988.
%K A205633 nonn,changed
%O A205633 0,1
%A A205633 _N. J. A. Sloane_, Feb 02 2012