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A131964 Expansion of f(x^2, x^10) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

%I A131964 #16 Feb 16 2025 08:33:06
%S A131964 1,1,1,2,0,1,1,0,1,0,2,1,1,1,0,1,2,2,0,1,1,1,1,1,0,1,1,0,1,0,1,2,1,1,
%T A131964 0,1,1,1,3,0,0,0,2,1,1,2,1,2,1,0,0,0,2,1,0,2,0,2,0,0,1,1,0,1,0,1,2,1,
%U A131964 2,1,1,1,1,0,0,0,2,1,2,0,2,2,1,1,0,0,1
%N A131964 Expansion of f(x^2, x^10) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.
%C A131964 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A131964 G. C. Greubel, <a href="/A131964/b131964.txt">Table of n, a(n) for n = 0..1000</a>
%H A131964 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A131964 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A131964 Expansion of phi(-x^4) * psi(-x^6) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.
%F A131964 Expansion of q^(-19/24) * eta(q^2) * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)) in powers of q.
%F A131964 Euler transform of period 24 sequence [ 1, 0, 1, -2, 1, -1, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, -1, 1, -2, 1, 0, 1, -2, ...].
%F A131964 a(25*n + 19) = a(n). a(25*n + 4) = a(25*n + 9) = a(25*n + 14) = a(25*n + 24) = 0.
%F A131964 2 * a(n) = A123484(24*n + 19).
%e A131964 G.f. = 1 + x + x^2 + 2*x^3 + x^5 + x^6 + x^8 + 2*x^10 + x^11 + x^12 + x^13 + ...
%e A131964 G.f. = q^19 + q^43 + q^67 + 2*q^91 + q^139 + q^163 + q^211 + 2*q^259 + q^283 + ...
%t A131964 a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 19}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* _Michael Somos_, Nov 03 2015 *)
%t A131964 a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/4) EllipticTheta[ 4, 0, x^4] QPochhammer[ -x, x] EllipticTheta[ 2, Pi/4, x^3], {x, 0, n}]; (* _Michael Somos_, Nov 03 2015 *)
%o A131964 (PARI) {a(n) = if( n<0, 0, n = 24*n + 19; sumdiv(n, d, kronecker( -12, d) * (n/d %2)) / 2)};
%o A131964 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)), n))};
%Y A131964 Cf. A123484.
%K A131964 nonn
%O A131964 0,4
%A A131964 _Michael Somos_, Aug 02 2007