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

0,2

Comments

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

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

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).

Example

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 + ...

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}];

Prog

(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))};

Crossrefs

Cf. A298932.

Sequence in context: A064558 A178031 A008328 * A365851 A091860 A333995

Adjacent sequences: A298930 A298931 A298932 * A298934 A298935 A298936

Keyword

nonn

Author

Michael Somos, Jan 29 2018

Status

approved