A108483 Expansion of f(-x^2, -x^5) / f(-x, -x^6) in powers of x where f() is a Ramanujan theta function.

1, 1, 0, 0, 0, -1, 0, 1, 1, 0, -1, -1, -1, 0, 2, 2, 0, -1, -2, -2, 0, 3, 3, 0, -2, -3, -3, 0, 5, 5, 1, -3, -5, -5, 0, 7, 7, 1, -5, -8, -7, 1, 11, 12, 2, -7, -12, -11, 1, 15, 16, 3, -11, -18, -15, 2, 23, 24, 5, -15, -26, -22, 3, 31, 33, 7, -22, -37, -30, 5, 44, 47, 11, -30, -52, -42, 6, 59, 63, 15, -42, -72, -56, 10, 82, 88, 22

Offset

0,15

Comments

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

In Duke (2005) page 157 the g.f. is denoted by t(tau).

References

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162.

Links

Table of n, a(n) for n=0..86.

Michael Somos, Introduction to Ramanujan theta functions [This is just section 1 of the next item, but it is mentioned in over 1000 sequences. - N. J. A. Sloane, Nov 13 2019]

Michael Somos, A Multisection of q-Series

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 7 sequence [ 1, -1, 0, 0, -1, 1, 0, ...]. - Michael Somos, Oct 03 2013

Given g.f. A(x), then B(q) = q^-2*A(q^7) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^3 - u^6 + 3*u^4*v + u^7*v^3 + u^2*v^9 + u^8*v^6 - 3*u^2*v^2 - 2*u*v^6 - 5*u^3*v^5 - u^5*v^4 - u^9*v^2 - u^4*v^8 - u^6*v^7.

G.f.: Product_{k>0} (1 - x^(7*k - 2)) * (1 - x^(7*k - 5)) / ((1 - x^(7*k - 1)) * (1 - x^(7*k - 6))).

a(n) = A229894(7*n). - Michael Somos, Oct 03 2013

Example

G.f. = 1 + x - x^5 + x^7 + x^8 - x^10 - x^11 - x^12 + 2*x^14 + 2*x^15 + ...
G.f. = q^-2 + q^5 - q^33 + q^47 + q^54 - q^68 - q^75 - q^82 + 2*q^96 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7] / (QPochhammer[ x, x^7] QPochhammer[ x^6, x^7]), {x, 0, n}]; (* Michael Somos, Oct 03 2013 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 1, 0, 0, 1, -1, 0}[[Mod[k, 7, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 03 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x*O(x^n))^[ 0, -1, 1, 0, 0, 1, -1][k%7 + 1]), n))};

Crossrefs

Cf. A229894.

Sequence in context: A058643 A343748 A029368 * A363623 A101565 A292944

Adjacent sequences: A108480 A108481 A108482 * A108484 A108485 A108486

Keyword

sign

Author

Michael Somos, Jun 04 2005

Status

approved