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.

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, 13, 12, 18, 8, 8, 12, 16, 14, 12, 12, 16, 12, 13, 14, 6, 20, 16, 18, 8, 10, 18, 16, 20, 12, 16, 16, 15, 20, 12, 18, 24, 14, 18, 8, 16, 18, 16, 22, 12, 12, 20, 24

Offset: 0

Author

Michael Somos, Jan 29 2018

Keywords

• nonn

Comments

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

Examples

			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 + ...
G.f. = q + 2*q^5 + 3*q^9 + 4*q^13 + 4*q^17 + 6*q^21 + 5*q^25 + 6*q^29 + ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A298932.

Programs

Maple

N:= 100:
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):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 29 2018

Mathematica

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}];

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))};

Formula

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.

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.

Euler transform of period 12 sequence [2, 0, 0, -1, 2, -3, 2, -1, 0, 0, 2, -3, ...].

a(n) = A298932(2*n).