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A281815 Expansion of f(x, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

%I A281815 #21 Feb 16 2025 08:33:40
%S A281815 1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,
%T A281815 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,
%U A281815 0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A281815 Expansion of f(x, x^10) in powers of x where f(, ) is Ramanujan's general theta function.
%H A281815 G. C. Greubel, <a href="/A281815/b281815.txt">Table of n, a(n) for n = 0..1000</a>
%H A281815 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A281815 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A281815 f(x,x^m) = 1 + Sum_{k>=1} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - _N. J. A. Sloane_, Jan 30 2017
%F A281815 Euler transform of period 22 sequence [1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, ...].
%F A281815 Characteristic function of generalized 13-gonal numbers A195313.
%F A281815 G.f.: Sum_{k in Z} x^(k*(11*k + 9)/2).
%F A281815 G.f.: Product_{k>0} (1 + x^(11*k-10)) * (1 + x^(11*k-1)) * (1 - x^(11*k)).
%F A281815 Sum_{k=1..n} a(k) ~ (2*sqrt(2/11)) * sqrt(n). - _Amiram Eldar_, Jan 13 2024
%e A281815 G.f. = 1 + x + x^10 + x^13 + x^31 + x^36 + x^63 + x^70 + x^106 + x^115 + ...
%e A281815 G.f. = q^81 + q^169 + q^961 + q^1225 + q^2809 + q^3249 + q^5625 + q^6241 + ...
%t A281815 a[ n_] := SquaresR[ 1, 88 n + 81] / 2;
%t A281815 a[ n_] := If[ n < 0, 0, Boole @ IntegerQ @ Sqrt @ (88 n + 81)];
%t A281815 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^11] QPochhammer[ -x^10, x^11] QPochhammer[ x^11], {x, 0, n}];
%o A281815 (PARI) {a(n) = issquare(88*n + 81)};
%Y A281815 Cf. f(x, x^k) for k=2..11: A080995, A010054, A133100, A089801, A274179, A214263, A281814, A205988, A281815, A186742.
%Y A281815 Cf. A195313.
%K A281815 nonn
%O A281815 0,1
%A A281815 _Michael Somos_, Jan 30 2017