A003823 Power series expansion of the Rogers-Ramanujan continued fraction 1+x/(1+x^2/(1+x^3/(1+x^4/(1+...)))).

1, 1, 0, -1, 0, 1, 1, -1, -2, 0, 2, 2, -1, -3, -1, 3, 3, -2, -5, -1, 6, 5, -3, -8, -2, 8, 7, -5, -12, -2, 13, 12, -7, -18, -4, 18, 16, -11, -26, -5, 27, 24, -14, -37, -8, 37, 33, -21, -52, -10, 53, 47, -29, -72, -15, 71, 63, -40, -98, -19, 99, 88, -53, -133, -27, 131, 115, -73, -178, -35, 177, 156, -95, -236, -48, 232, 204, -127, -311

Offset

0,9

Comments

This is the q-expansion of the Gamma(5)-modular function (or automorphic function) Lambda given, for example, in Erdelyi et al., Higher Transcendental Functions eq. 44 volume 3 page 24 sec. 14.6.3 - Warren Smith.

Number 14 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(5). [Yang 2004] - Michael Somos, Aug 07 2014

References

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 81.

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 24.

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 404.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

G. E. Andrews, Ramanujan's "lost" notebook, III, the Rogers-Ramanujan continued fraction, Adv. Math. 41 (1981), 186-208.

S.-D. Chen and S.-S. Huang, On the series expansion of the Göllnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eq. (6.5).

J. Malenfant, Generalizing Ramanujan's J Functions, arXiv preprint arXiv:1109.5957 [math.NT], 2011.

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

G.f.: Prod_{k>0} (1-x^{5k-2})(1-x^{5k-3})/((1-x^{5k-1})(1-x^{5k-4})).

G.f.: (Sum_{k in Z} (-1)^k * x^((5*k + 1) * k/2)) / (Sum_{k in Z} (-1)^k * x^((5*k + 3) * k/2)). - Michael Somos, Dec 13 2002

Euler transform of period 5 sequence [1, -1, -1, 1, 0, ...]. - Michael Somos, Dec 13 2002

G.f. is reciprocal of that for the Rogers-Ramanujan continued fraction r(tau) - see A007325.

Expansion of f(-x^2, -x^3) / f(-x, -x^4) in powers of x where f(,) is Ramanujan's two-variable theta function. - Michael Somos, Aug 07 2014

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109091(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 01 2017

Example

G.f. = 1 + x - x^3 + x^5 + x^6 - x^7 - 2*x^8 + 2*x^10 + 2*x^11 - x^12 - ...
G.f. = 1/q + q^4 - q^14 + q^24 + q^29 - q^34 - 2*q^39 + 2*q^49 + 2*q^54 - q^59 + ...

Maple

M := 100: a[ M ] := 1+z; for n from M-1 by -1 to 1 do a[ n ] := series( 1 + z^n/a[ n+1 ], z, M+1); od: a[ 1 ];
M:=100; qf:=(a, q)->mul(1-a*q^j, j=0..M); t1:=qf(q^2, q^5)*qf(q^3, q^5)/(qf(q, q^5)*qf(q^4, q^5)); series(%, q, M); seriestolist(%);

Mathematica

kmax = 16; f[x_] := Product[(1-x^(5k-2))*(1-x^(5k-3))/((1-x^(5k-1))*(1-x^(5k-4))), {k, 1, kmax}]; CoefficientList[ Series[f[x], {x, 0, 5*kmax}], x] (* Jean-François Alcover, Nov 02 2011, after g.f. *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, Jul 09 2014 *)
a[ n_] := If[n < 0, 0, SeriesCoefficient[ 1 / ContinuedFractionK[ x^k, 1, {k, 0, n}], {x, 0, n}]]; (* Michael Somos, Jul 09 2014 *)

Prog

(PARI) {a(n) = local(k); if( n<0, 0, k = (3 + sqrtint(9 + 40*n)) \ 10; polcoeff( sum( i=-k, k, (-1)^i * x^((5*i^2 + i)/2), x * O(x^n)) / sum( i=-k, k, (-1)^i * x^((5*i^2 + 3*i)/2), x * O(x^n)), n))}; /* Michael Somos, Dec 13 2002 */
(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, if( k%5, (1 - x^k)^( -(-1)^binomial( k%5, 2)), 1), 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 13 2002 */
(PARI) {a(n) = local(cf); if( n<0, 0, cf = contfracpnqn( matrix(2, (sqrtint(8*n + 1) + 1)\2, i, j, if( i==1, x^(j-1), 1))); polcoeff( cf[1, 1] / cf[2, 1] + x * O(x^n), n))}; /* Michael Somos, Dec 13 2002 */

Crossrefs

Cf. A007325.

Sequence in context: A308626 A268755 A128664 * A059451 A083817 A286222

Adjacent sequences: A003820 A003821 A003822 * A003824 A003825 A003826

Keyword

sign,easy,nice

Author

N. J. A. Sloane

Status

approved