A263074 Expansion of phi(-x) / (chi(-x^3) * chi(-x^5)) in powers of x where phi(), chi() are Ramanujan theta functions.

1, -2, 0, 1, 0, 1, -1, 0, 1, 0, -1, -1, 0, 1, 0, 1, -3, 0, 2, 0, 1, -1, 0, 2, 0, 0, -3, 0, 1, 0, 2, -4, 0, 2, 0, 1, -3, 0, 3, 0, 1, -4, 0, 2, 0, 3, -6, 0, 4, 0, 4, -6, 0, 4, 0, 1, -7, 0, 4, 0, 3, -9, 0, 5, 0, 4, -8, 0, 6, 0, 3, -10, 0, 6, 0, 6, -13, 0, 8, 0, 5

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^(-1/3) * eta(q)^2 * eta(q^6) * eta(q^10) / (eta(q^2) * eta(q^3) * eta(q^5)) in powers of q.

Euler transform of period 30 sequence [ -2, -1, -1, -1, -1, -1, -2, -1, -1, -1, -2, -1, -2, -1, 0, -1, -2, -1, -2, -1, -1, -1, -2, -1, -1, -1, -1, -1, -2, -1, ...].

G.f.: Product_{k>0} (1 + x^(3*k)) * (1 + x^(5*k)) * (1 - x^k) / (1 + x^k).

Convolution inverse is A100823.

a(5*n + 2) = a(5*n + 4) = 0. a(5*n + 3) = A263073(n).

Example

G.f. = 1 - 2*x + x^3 + x^5 - x^6 + x^8 - x^10 - x^11 + x^13 + x^15 + ...
G.f. = q - 2*q^4 + q^10 + q^16 - q^19 + q^25 - q^31 - q^34 + q^40 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / (QPochhammer[ x^3, x^6] QPochhammer[ x^5, x^10]), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A)), n))};

Crossrefs

Cf. A100823, A263073.

Sequence in context: A319660 A285726 A285005 * A281772 A082886 A287179

Adjacent sequences: A263071 A263072 A263073 * A263075 A263076 A263077

Keyword

sign

Author

Michael Somos, Oct 08 2015

Status

approved