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

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

Offset: 0

Author

Michael Somos, Jan 30 2017

Keywords

• nonn

Examples

			G.f. = 1 + x + x^10 + x^13 + x^31 + x^36 + x^63 + x^70 + x^106 + x^115 + ...
G.f. = q^81 + q^169 + q^961 + q^1225 + q^2809 + q^3249 + q^5625 + q^6241 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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. A195313.

Programs

Mathematica

a[ n_] := SquaresR[ 1, 88 n + 81] / 2;
a[ n_] := If[ n < 0, 0, Boole @ IntegerQ @ Sqrt @ (88 n + 81)];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^11] QPochhammer[ -x^10, x^11] QPochhammer[ x^11], {x, 0, n}];

PARI

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

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 22 sequence [1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, ...].

Characteristic function of generalized 13-gonal numbers A195313.

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

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

Sum_{k=1..n} a(k) ~ (2*sqrt(2/11)) * sqrt(n). - Amiram Eldar, Jan 13 2024