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A186741 Expansion of f(x^5, x^7) in powers of x where f(, ) is Ramanujan's general theta function.

%I A186741 #27 Jul 13 2025 21:01:22
%S A186741 1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,
%T A186741 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
%U A186741 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0
%N A186741 Expansion of f(x^5, x^7) in powers of x where f(, ) is Ramanujan's general theta function.
%C A186741 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A186741 Seiichi Manyama, <a href="/A186741/b186741.txt">Table of n, a(n) for n = 0..1000</a>
%H A186741 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A186741 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A186741 Euler transform of period 24 sequence [ 0, 0, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, ...].
%F A186741 a(n) is the characteristic function of A036498. a(n) = max( 0, A010815(n)).
%F A186741 G.f.: Sum_{k in Z} x^(6*k^2 - k) = Product_{k>0} (1 + x^(12*k - 7)) * (1 + x^(12*k - 5)) * (1 - x^(12*k)).
%F A186741 Sum_{k=1..n} a(k) ~ sqrt(2*n/3). - _Amiram Eldar_, Jan 13 2024
%e A186741 G.f. = 1 + x^5 + x^7 + x^22 + x^26 + x^51 + x^57 + x^92 + x^100 + x^145 + ...
%e A186741 G.f. = q + q^121 + q^169 + q^529 + q^625 + q^1225 + q^1369 + q^2209 + q^2401 + ...
%t A186741 a[n_]:= SeriesCoefficient[ QPochhammer[-q^5,q^12]*QPochhammer[-q^7,q^12] *QPochhammer[q^12,q^12], {q, 0, n}]; (* _G. C. Greubel_, Dec 08 2017 *)
%o A186741 (PARI) {a(n) = my(m); if( !issquare( 24*n + 1, &m), 0, m%12 == 1 || m%12 == 11)};
%Y A186741 Cf. A010815, A036498, A247223.
%Y A186741 Cf. A000122, A000700, A010054, A121373.
%K A186741 nonn
%O A186741 0,1
%A A186741 _Michael Somos_, Jan 21 2012