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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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 * A091860 A301764 A181833

Adjacent sequences:  A298930 A298931 A298932 * A298934 A298935 A298936

KEYWORD

nonn

AUTHOR

Michael Somos, Jan 29 2018

STATUS

approved

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Last modified March 28 16:48 EDT 2020. Contains 333089 sequences. (Running on oeis4.)