A212770 Expansion of q / (chi(q) * chi(q^2) * chi(q^3) * chi(q^6))^2 in powers of q where chi() is a Ramanujan theta function.

1, -2, 1, -4, 10, -10, 12, -24, 37, -44, 56, -84, 126, -160, 186, -272, 394, -466, 568, -792, 1052, -1272, 1560, -2040, 2663, -3244, 3877, -4992, 6410, -7644, 9180, -11616, 14472, -17284, 20712, -25572, 31518, -37576, 44510, -54416, 66402, -78368, 92648

Offset

1,2

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 = 1..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of b(q) * c(q) * b(q^8) * c(q^8) / (b(q^2) * c(q^2) * b(q^4) * c(q^4)) in powers of q where b(), c() are cubic AGM theta functions.

Expansion of (eta(q) * eta(q^3) * eta(q^8) * eta(q^24) / (eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12)))^2 in powers of q.

Euler transform of period 24 sequence [ -2, 0, -4, 2, -2, 0, -2, 0, -4, 0, -2, 4, -2, 0, -4, 0, -2, 0, -2, 2, -4, 0, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t).

a(2*n) = -2 * A123647(n). a(4*n) = -4 * A123653(n).

Example

G.f. = x - 2*x^2 + x^3 - 4*x^4 + 10*x^5 - 10*x^6 + 12*x^7 - 24*x^8 + 37*x^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, -q] QPochhammer[ q^2, -q^2] QPochhammer[ q^3, -q^3] QPochhammer[ q^6, -q^6])^2, {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n = n-1; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^24 + A) / (eta(x^2 + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^12 + A)))^2, n))};

Crossrefs

Cf. A123647, A123653.

Sequence in context: A179077 A352093 A038195 * A205855 A329709 A038521

Adjacent sequences: A212767 A212768 A212769 * A212771 A212772 A212773

Keyword

sign

Author

Michael Somos, May 26 2012

Status

approved