A281815 - OEIS

%I #20 Jan 13 2024 03:39:30

%S 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 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 0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

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

%H G. C. Greubel, <a href="/A281815/b281815.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F 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 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 Characteristic function of generalized 13-gonal numbers A195313.

%F G.f.: Sum_{k in Z} x^(k*(11*k + 9)/2).

%F G.f.: Product_{k>0} (1 + x^(11*k-10)) * (1 + x^(11*k-1)) * (1 - x^(11*k)).

%F Sum_{k=1..n} a(k) ~ (2*sqrt(2/11)) * sqrt(n). - _Amiram Eldar_, Jan 13 2024

%e G.f. = 1 + x + x^10 + x^13 + x^31 + x^36 + x^63 + x^70 + x^106 + x^115 + ...

%e G.f. = q^81 + q^169 + q^961 + q^1225 + q^2809 + q^3249 + q^5625 + q^6241 + ...

%t a[ n_] := SquaresR[ 1, 88 n + 81] / 2;

%t a[ n_] := If[ n < 0, 0, Boole @ IntegerQ @ Sqrt @ (88 n + 81)];

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^11] QPochhammer[ -x^10, x^11] QPochhammer[ x^11], {x, 0, n}];

%o (PARI) {a(n) = issquare(88*n + 81)};

%Y Cf. f(x, x^k) for k=2..11: A080995, A010054, A133100, A089801, A274179, A214263, A281814, A205988, A281815, A186742.

%Y Cf. A195313.

%K nonn

%O 0,1

%A _Michael Somos_, Jan 30 2017