cp's OEIS Frontend

A204843 Expansion of x * f(-x^24) + (3 * phi(-x^36) - phi(-x^4)) / 2 in powers of x where phi(), f() are Ramanujan theta functions.

%I A204843 #18 Feb 16 2025 08:33:16
%S A204843 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 A204843 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 A204843 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 A204843 Expansion of x * f(-x^24) + (3 * phi(-x^36) - phi(-x^4)) / 2 in powers of x where phi(), f() are Ramanujan theta functions.
%C A204843 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A204843 Antti Karttunen, <a href="/A204843/b204843.txt">Table of n, a(n) for n = 0..65537</a>
%H A204843 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A204843 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A204843 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 A204843 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 A204843 Euler transform of period 24 sequence [ 1, -1, 0, 1, -1, 1, -1, 0, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, -1, 1, 0, -1, 1, -1, ...].
%F A204843 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) = A204853(n).
%e A204843 1 + x + x^4 - x^16 - x^25 - 2*x^36 - x^49 - x^64 + x^100 + x^121 + ...
%t A204843 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 A204843 (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 A204843 Cf. A089810, A010815, A204853.
%K A204843 sign
%O A204843 0,37
%A A204843 _Michael Somos_, Jan 19 2012