A257628 Expansion of 1 - f(-x) in powers of x where f() is a Ramanujan theta function.

0, 1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 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

Offset

0

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Wikipedia, Pentagonal number theorem

Formula

G.f.: x + x^2 * (1 - x) + x^3 * (1 - x) * (1 - x^2) + ....

G.f.: Sum_{k>0} -(-1)^k * (x^((3*k^2 - k)/2) + x^((3*k^2 + k)/2)).

G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k) / 2) / ((1 - x) * (1 - x^2) * ... * (1 - x^k)).

G.f.: -(Product_{j>=1}(1-x^j) - 1), from Euler's Pentagonal Theorem. - Wolfdieter Lang, Feb 16 2021

a(n) = - A010815(n) unless n=0, a(0) = 0.

Example

G.f. = x + x^2 - x^5 - x^7 + x^12 + x^15 - x^22 - x^26 + x^35 + x^40 + ...
G.f. = q^25 + q^49 - q^121 - q^169 + q^289 + q^361 - q^529 - q^625 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1 - QPochhammer[ x], {x, 0, n}];
a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ n > 0 && IntegerQ[m], - KroneckerSymbol[ 12, m], 0]];

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 - eta(x + x * O(x^n)), n))};
(PARI) {a(n) = my(m); if( n>0 && issquare( 24*n + 1, &m), - kronecker( 12, m))};

Crossrefs

Cf. A001318, A010815, A341418 (convolution triangle).

Sequence in context: A288752 A189706 A188321 * A203568 A327974 A286049

Adjacent sequences: A257625 A257626 A257627 * A257629 A257630 A257631

Keyword

sign,easy

Author

Michael Somos, Jul 12 2015

Status

approved