A204853 - OEIS

%I #12 Mar 12 2021 22:24:46

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

%T 0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,

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

%N Expansion of (3 * phi(-x^36) - phi(-x^4)) / 2 - x * f(-x^24) in powers of x where phi(), f() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A204853/b204853.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 Expansion of phi(-x^36) - x * f(-x^24) + x^4 * f(-x^12, -x^60) in powers of x where phi(), f() are Ramanujan theta functions.

%F Expansion of f(-x^9, x^9) - x * f(-x^3, x^15) in powers of x where f() is the two variable Ramanujan theta function.

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

%F a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 3) = a(6*n + 3) = a(8*n + 5) =  0. a(4*n) = A089810(n). a(24*n + 1) = - A010815(n). a(25*n) = a(49*n) = A204843(n). a(n) = (-1)^n * A204843(n).

%e 1 - x + x^4 - x^16 + x^25 - 2*x^36 + x^49 - x^64 + x^100 - x^121 + ...

%t a[n_]:= SeriesCoefficient[(3*EllipticTheta[3, 0, -q^36] -EllipticTheta[3, 0, -q^4])/2 - q*QPochhammer[q^24, q^72]*QPochhammer[q^48, q^72]* QPochhammer[q^72, q^72], {q, 0, n}]; Table[a[n], {n,0,100}] (* _G. C. Greubel_, Dec 19 2017 *)

%o (PARI) {a(n) = local(m); if( n<1, n==0, if( issquare( n, &m), (-1)^(m\6) * [ 2, -1, 1, 0, -1, 1][m%6 + 1]))}

%Y Cf. A010815, A089810, A204843.

%K sign

%O 0,37

%A _Michael Somos_, Jan 20 2012