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A214639 Expansion of q * f(-q^2, -q^14) / f(-q, q^3) in powers of q where f(,) is Ramanujan's two-variable theta function.

%I A214639 #24 Feb 16 2025 08:33:18
%S A214639 0,1,1,0,-1,-2,-2,0,3,5,4,0,-6,-10,-8,0,11,18,15,0,-20,-32,-26,0,34,
%T A214639 55,44,0,-56,-90,-72,0,91,144,114,0,-143,-226,-178,0,220,346,272,0,
%U A214639 -334,-522,-408,0,498,777,605,0,-732,-1138,-884,0,1064,1648,1276,0
%N A214639 Expansion of q * f(-q^2, -q^14) / f(-q, q^3) in powers of q where f(,) is Ramanujan's two-variable theta function.
%C A214639 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A214639 G. C. Greubel, <a href="/A214639/b214639.txt">Table of n, a(n) for n = 0..1000</a>
%H A214639 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214639 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214639 Expansion of q * f(q, q^7)^2 / (phi(q^4) * psi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A214639 Euler transform of period 16 sequence [ 1, -1, -1, -1, -1, 1, 1, 2, 1, 1, -1, -1, -1, -1, 1, 0, ...].
%F A214639 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 2 * u * v * (1 - u - u*v).
%F A214639 a(4*n + 3) = 0. a(4*n) = (-1)^n * A185083(n) unless n=0. a(4*n + 1) = A079006(n). a(4*n + 2) = A210063(n).
%F A214639 a(2*n) = A224216(n).
%e A214639 q + q^2 - q^4 - 2*q^5 - 2*q^6 + 3*q^8 + 5*q^9 + 4*q^10 - 6*q^12 - 10*q^13 + ...
%t A214639 nn = 16*10; b = Flatten[Table[{1, -1, -1, -1, -1, 1, 1, 2, 1, 1, -1, -1, -1, -1, 1, 0}, {nn/16}]]; CoefficientList[x * Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* _T. D. Noe_, Jul 25 2012 *)
%t A214639 a[ n_] :=  SeriesCoefficient[ 2 q^(9/8) QPochhammer[ -q, q^8]^2 QPochhammer[ q^8]^2 QPochhammer[ -q^7, q^8]^2 / (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 3, 0, q^4]), {q, 0, n}] (* _Michael Somos_, Apr 02 2013 *)
%o A214639 (PARI) {a(n) = local(A); if( n<1, 0, if( n%4==3, 0, if( n%2==0, n/=2; A = x * O(x^n); (-1)^(n\2) * polcoeff( eta(-x + A) / eta(x + A), n) / 2, n\=4; A = x * O(x^n); (-1)^n * polcoeff( (eta(x^4 + A) / eta(x + A))^2, n))))}
%o A214639 (PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod( k=1, n, (1 -x^k + x * O(x^n))^[ 0, -1, 1, 1, 1, 1, -1, -1, -2, -1, -1, 1, 1, 1, 1, -1][k%16 + 1]), n))}
%Y A214639 Cf. A079006, A185083, A210063, A214263, A224216.
%K A214639 sign
%O A214639 0,6
%A A214639 _Michael Somos_, Jul 24 2012