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A298933 Expansion of f(x, x^2) * f(x, x^3) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

%I A298933 #12 Feb 16 2025 08:33:53
%S A298933 1,2,3,4,4,6,5,6,6,4,8,6,9,6,6,12,8,12,8,8,9,8,12,6,8,14,12,12,8,12,
%T A298933 13,12,18,8,8,12,16,14,12,12,16,12,13,14,6,20,16,18,8,10,18,16,20,12,
%U A298933 16,16,15,20,12,18,24,14,18,8,16,18,16,22,12,12,20,24
%N A298933 Expansion of f(x, x^2) * f(x, x^3) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general theta function.
%C A298933 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A298933 Robert Israel, <a href="/A298933/b298933.txt">Table of n, a(n) for n = 0..10000</a>
%H A298933 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A298933 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A298933 Expansion of phi(x) * phi(-x^3) * phi(-x^6) / chi(-x^2)^3 in powers of x where phi(), chi() are Ramanujan theta functions.
%F A298933 Expansion of q^(-1/4) * eta(q^2)^2 * eta(q^3)^2 * eta(q^4) * eta(q^6) / (eta(q)^2 * eta(q^12)) in powers of q.
%F A298933 Euler transform of period 12 sequence [2, 0, 0, -1, 2, -3, 2, -1, 0, 0, 2, -3, ...].
%F A298933 a(n) = A298932(2*n).
%e A298933 G.f. = 1 + 2*x + 3*x^2 + 4*x^3 + 4*x^4 + 6*x^5 + 5*x^6 + 6*x^7 + 6*x^8 + ...
%e A298933 G.f. = q + 2*q^5 + 3*q^9 + 4*q^13 + 4*q^17 + 6*q^21 + 5*q^25 + 6*q^29 + ...
%p A298933 N:= 100:
%p A298933 S:= series(JacobiTheta3(0,x)*JacobiTheta4(0,x^3)*JacobiTheta4(0,x^6)*expand(QDifferenceEquations:-QPochhammer(-x^2,x^2,floor(N/2)))^3, x, N+1):
%p A298933 seq(coeff(S,x,j),j=0..N); # _Robert Israel_, Jan 29 2018
%t A298933 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^2]^3, {x, 0, n}];
%o A298933 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^6 + A) / (eta(x + A)^2 * eta(x^12 + A)), n))};
%Y A298933 Cf. A298932.
%K A298933 nonn
%O A298933 0,2
%A A298933 _Michael Somos_, Jan 29 2018