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A281814 Expansion of f(x, x^8) in powers of x where f(, ) is Ramanujan's general theta function.

%I A281814 #20 Feb 16 2025 08:33:40
%S A281814 1,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,
%T A281814 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
%U A281814 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A281814 Expansion of f(x, x^8) in powers of x where f(, ) is Ramanujan's general theta function.
%H A281814 G. C. Greubel, <a href="/A281814/b281814.txt">Table of n, a(n) for n = 0..5000</a>
%H A281814 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A281814 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A281814 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 A281814 Euler transform of period 18 sequence [1, -1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, -1, 1, -1, ...].
%F A281814 Characteristic function of generalized 11-gonal numbers A195160.
%F A281814 G.f.: Sum_{k in Z} x^(k*(9*k + 7)/2).
%F A281814 G.f.: Product_{k>0} (1 + x^(9*k-8)) * (1 + x^(9*k-1)) * (1 - x^(9*k)).
%F A281814 Sum_{k=1..n} a(k) ~ (2*sqrt(2)/3) * sqrt(n). - _Amiram Eldar_, Jan 13 2024
%e A281814 G.f. = 1 + x + x^8 + x^11 + x^25 + x^30 + x^51 + x^58 + x^86 + x^95 + ...
%e A281814 G.f. = q^49 + q^121 + q^625 + q^841 + q^1849 + q^2209 + q^3721 + q^4225 + ...
%t A281814 a[ n_] := SquaresR[ 1, 72 n + 49] / 2;
%t A281814 a[ n_] := If[ n < 0, 0, Boole @ IntegerQ @ Sqrt @ (72 n + 49)];
%t A281814 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^9] QPochhammer[ -x^8, x^9] QPochhammer[ x^9], {x, 0, n}];
%o A281814 (PARI) {a(n) = issquare(72*n + 49)};
%Y A281814 Cf. f(x, x^k) for k=2..11: A080995, A010054, A133100, A089801, A274179, A214263, A281814, A205988, A281815, A186742.
%Y A281814 Cf. A195160, A281490.
%K A281814 nonn
%O A281814 0,1
%A A281814 _Michael Somos_, Jan 30 2017