A185124 - OEIS

%I #21 Oct 18 2024 20:14:48

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

%T 0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0

%N Expansion of f(x, -x^5) in powers of x where f(,) is the Ramanujan general theta function.

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

%C a(n) is nonzero if and only if n is a number of A001082.

%C The exponents in the q-series for this sequence are the squares of the numbers of A001651.

%H G. C. Greubel, <a href="/A185124/b185124.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>

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

%F Euler transform of period 24 sequence [ 1, -1, 0, 0, -1, 1, -1, 0, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, -1, 0, 0, -1, 1, -1, ...].

%F G.f.: Sum_{k in Z} (-1)^floor((k + 1)/2) * x^(k * (3*k + 2)).

%F a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 4) = a(8*n + 4) = 0. a(4*n + 1) = A080902(n). a(8*n) = A010815(n).

%F a(n) = (-1)^n * A185125(n). - _Michael Somos_, Jun 30 2015

%e G.f. = 1 + x - x^5 - x^8 - x^16 - x^21 + x^33 + x^40 + x^56 + x^65 - x^85 + ...

%e G.f. = q + q^4 - q^16 - q^25 - q^49 - q^64 + q^100 + q^121 + q^169 + q^196 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, -x^6] QPochhammer[ x^5, -x^6] QPochhammer[ -x^6], {x, 0, n}]; (* _Michael Somos_, Jun 30 2015 *)

%o (PARI) {a(n) = my(m); if( issquare( 3*n + 1, &m), (m%3!=0) * (-1)^((m+3) \ 6), 0)};

%Y Cf. A010815, A080902, A185125.

%K sign,easy

%O 0,1

%A _Michael Somos_, Jan 20 2012