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A186742 Expansion of f(x, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

%I A186742 #25 Feb 16 2025 08:33:14
%S A186742 1,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A186742 1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A186742 0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A186742 Expansion of f(x, x^11) in powers of x where f(, ) is Ramanujan's general theta function.
%C A186742 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A186742 Seiichi Manyama, <a href="/A186742/b186742.txt">Table of n, a(n) for n = 0..10000</a>
%H A186742 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A186742 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A186742 Euler transform of period 24 sequence [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, ...].
%F A186742 a(n) is the characteristic function of A195818. a(n) = max( 0, -A010815(n + 1)).
%F A186742 G.f.: Sum_{k in Z} x^(6*k^2 - 5*k) = Product_{k>0} (1 + x^(12*k - 11)) * (1 + x^(12*k - 1)) * (1 - x^(12*k)).
%F A186742 Expansion of (f(x, x^2) - f(-x, -x^2)) / (2*x) in powers of x. - _Michael Somos_, Aug 28 2017
%F A186742 Sum_{k=1..n} a(k) ~ sqrt(2*n/3). - _Amiram Eldar_, Jan 13 2024
%e A186742 G.f. = 1 + x + x^11 + x^14 + x^34 + x^39 + x^69 + x^76 + x^116 + x^125 + ...
%e A186742 G.f. = q^25 + q^49 + q^289 + q^361 + q^841 + q^961 + q^1681 + q^1849 + ...
%t A186742 a[ n_] := With[{m = Sqrt[24 n + 25]}, If[ n >= 0 && IntegerQ @ m, Boole[ Mod[m, 12] == 5 || Mod[m, 12] == 7], 0]]; (* _Michael Somos_, Aug 28 2017 *)
%t A186742 a[ n_] := SeriesCoefficient[ QPochhammer[ x^12] QPochhammer[ -x^1, x^12] QPochhammer[ -x^11, x^12], {x, 0, n}]; (* _Michael Somos_, Aug 28 2017 *)
%o A186742 (PARI) {a(n) = my(m); n++; if( !issquare( 24*n + 1, &m), 0, m%12 == 5 || m%12 == 7)};
%o A186742 (PARI) {a(n) = my(A); n = 3*n + 3; if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A) - eta(x^3 + A) - x * eta(x^18 + A)^2 / eta(x^9 + A)) / 2, n))}; /* _Michael Somos_, Aug 28 2017 */
%Y A186742 Cf. A010815, A195818.
%Y A186742 Cf. A000122, A000700, A010054, A121373.
%K A186742 nonn
%O A186742 0,1
%A A186742 _Michael Somos_, Jan 21 2012