A133100 Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset

0,1

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for characteristic functions.

Formula

f(x,x^m) = 1 + Sum_{k=1..oo} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - N. J. A. Sloane, Jan 30 2017

The characteristic function of A085787 generalized heptagonal numbers.

Euler transform of period 10 sequence [1, -1, 0, 1, -1, 1, 0, -1, 1, -1, ...].

G.f.: Prod_{k>0} (1 - x^(5*k)) * (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)) = Sum_{k in Z} x^((5*k^2 + 3*k) / 2).

a(n) = |A113429(n)|. a(3*n + 2) = 0.

Sum_{k=1..n} a(k) ~ 2 * sqrt(2/5) * sqrt(n). - Amiram Eldar, Jan 13 2024

Example

G.f. = 1 + x + x^4 + x^7 + x^13 + x^18 + x^27 + x^34 + x^46 + x^55 + x^70 + ...
G.f. = q^9 + q^49 + q^169 + q^289 + q^529 + q^729 + q^1089 + q^1369 + q^1849 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^5] QPochhammer[ -x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := SquaresR[ 1, 40 n + 9] / 2; (* Michael Somos, Jan 30 2017 *)
a[ n_] := If[n < 0, 0, Boole @ IntegerQ @ Sqrt @ (40 n + 9)]; (* Michael Somos, Jan 30 2017 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k*[-1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))};
(PARI) {a(n) = issquare( 40*n + 9)};

Crossrefs

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

Cf. A085787, A113429.

Sequence in context: A186447 A118009 A113429 * A216230 A077606 A004601

Adjacent sequences: A133097 A133098 A133099 * A133101 A133102 A133103

Keyword

nonn

Author

Michael Somos, Sep 11 2007

Status

approved