A214341 Expansion of 1 / k(q) = 1 / (r(q) * r(q^2)^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.

1, 1, 2, 1, 1, 0, -1, -2, -2, -1, 1, 3, 4, 4, 1, -2, -6, -8, -7, -3, 4, 10, 14, 12, 6, -6, -16, -22, -20, -8, 8, 26, 34, 31, 12, -14, -41, -54, -47, -20, 23, 61, 84, 72, 31, -32, -90, -122, -107, -44, 45, 133, 174, 154, 61, -68, -192, -254, -220, -90, 100

Offset

-1,3

Comments

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

Number 12 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Aug 07 2014

A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(10). [Yang 2004] - Michael Somos, Aug 07 2014

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

Formula

Expansion of (1/x) * (f(-x^4, -x^6) * f(-x^3, -x^7)) / (f(-x^2, -x^8) * f(-x, -x^9)) in powers of x where f(,) is Ramanujan's two-variable theta function.

Euler transform of period 10 sequence [ 1, 1, -1, -1, 0, -1, -1, 1, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^2 - v * (u^2 - 1).

G.f.: (1/x) * Product_{k>0} (1 - x^(10*k - 3)) * (1 - x^(10*k - 4)) * (1 - x^(10*k - 6)) * (1 - x^(10*k - 7)) /((1 - x^(10*k - 1)) * (1 - x^(10*k - 2)) * (1 - x^(10*k - 8)) * (1 - x^(10*k - 9))).

Convolution inverse of A112274. a(n) = A112274(n) + A132980(n).

Example

G.f. = 1/q + 1 + 2*q + q^2 + q^3 - q^5 - 2*q^6 - 2*q^7 - q^8 + q^9 + 3*q^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q Product[(1 - q^k)^{-1, -1, 1, 1, 0, 1, 1, -1, -1, 0}[[Mod[k, 10, 1]]], {k, n + 1}], {q, 0, n}]; (* Michael Somos, Aug 07 2014 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( prod( k=1, n, (1 - x^k + A)^[0, -1, -1, 1, 1, 0, 1, 1, -1, -1][k%10 + 1]), n))};

Crossrefs

Cf. A112274, A132980.

Sequence in context: A277264 A259538 A099314 * A281871 A131334 A004602

Adjacent sequences: A214338 A214339 A214340 * A214342 A214343 A214344

Keyword

sign

Author

Michael Somos, Jul 12 2012

Status

approved