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

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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Michael Somos, Jan 30 2017

Keywords

• nonn

Examples

			G.f. = 1 + x + x^8 + x^11 + x^25 + x^30 + x^51 + x^58 + x^86 + x^95 + ...
G.f. = q^49 + q^121 + q^625 + q^841 + q^1849 + q^2209 + q^3721 + q^4225 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

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

Cf. A195160, A281490.

Programs

Mathematica

a[ n_] := SquaresR[ 1, 72 n + 49] / 2;
a[ n_] := If[ n < 0, 0, Boole @ IntegerQ @ Sqrt @ (72 n + 49)];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^9] QPochhammer[ -x^8, x^9] QPochhammer[ x^9], {x, 0, n}];

PARI

{a(n) = issquare(72*n + 49)};

Formula

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

Euler transform of period 18 sequence [1, -1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, -1, 1, -1, ...].

Characteristic function of generalized 11-gonal numbers A195160.

G.f.: Sum_{k in Z} x^(k*(9*k + 7)/2).

G.f.: Product_{k>0} (1 + x^(9*k-8)) * (1 + x^(9*k-1)) * (1 - x^(9*k)).

Sum_{k=1..n} a(k) ~ (2*sqrt(2)/3) * sqrt(n). - Amiram Eldar, Jan 13 2024