A263021 Expansion of f(-x^3)^6 / (phi(-x) * phi(-x^3)) in powers of x where phi(), f() are Ramanujan theta functions.

1, 2, 4, 4, 6, 8, 9, 10, 8, 14, 14, 16, 16, 16, 20, 18, 22, 24, 21, 26, 28, 28, 28, 24, 36, 34, 36, 38, 32, 32, 40, 42, 44, 36, 46, 56, 43, 50, 40, 52, 54, 56, 54, 42, 60, 62, 64, 64, 56, 66, 56, 72, 70, 56, 74, 74, 76, 72, 64, 80, 81, 84, 84, 64, 76, 88, 88

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-3/4) * eta(q^2) * eta(q^3)^4 * eta(q^6) / eta(q)^2 in powers of q.

Euler transform of period 6 sequence [ 2, 1, -2, 1, 2, -4, ...].

a(3*n) = A261445(n). a(3*n + 1) = 2 * A260518(n). a(3*n + 2) = 4 * A260295(n).

Example

G.f. = 1 + 2*x + 4*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 9*x^6 + 10*x^7 + 8*x^8 + ...
G.f. = q^3 + 2*q^7 + 4*q^11 + 4*q^15 + 6*q^19 + 8*q^23 + 9*q^27 + 10*q^31 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^6 / (EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3]), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^4 * eta(x^6 + A) / eta(x + A)^2, n))};

Crossrefs

Cf. A260295, A260518, A261445.

Sequence in context: A000061 A153176 A229144 * A112921 A339905 A215205

Adjacent sequences: A263018 A263019 A263020 * A263022 A263023 A263024

Keyword

nonn

Author

Michael Somos, Oct 07 2015

Status

approved