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A003823
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Power series expansion of the Rogers-Ramanujan continued fraction 1+x/(1+x^2/(1+x^3/(1+x^4/(1+...)))).
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7
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| This is the q-expansion of the Gamma(5)-modular function (or automorphic function) Lambda given for example in Erdelyi et al., Higher Transc. Fns. eq. 44 volume 3 page 24 sec. 14.6.3 - Warren Smith.
Euler transform of period 5 sequence [1,-1,-1,1,0,...].
G.f. is reciprocal of that for the Rogers-Ramanujan continued fraction r(tau) - see A007325.
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REFERENCES
| G. E. Andrews, Ramanujan's "lost" notebook, III, the Rogers-Ramanujan continued fraction, Adv. Math. 41 (1981), 186-208.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 81.
S.-D. Chen and S.-S. Huang, On the series expansion of the Goellnitz-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).
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 24.
J. Malenfant, Generalizing Ramanujan's J Functions, Arxiv preprint arXiv:1109.5957, 2011
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 404.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
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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_{-oo..oo} (-1)^n x^((5n+1)n/2))/(Sum_{-oo..oo} (-1)^n x^((5n+3)n/2)). - Michael Somos, Dec 13 2002
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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(%);
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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] (* From Jean-François Alcover, Nov 02 2011, after g.f. *)
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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))
(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))
(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))
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CROSSREFS
| Cf. A007325.
Sequence in context: A071635 A156643 A128664 * A059451 A083817 A029273
Adjacent sequences: A003820 A003821 A003822 * A003824 A003825 A003826
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KEYWORD
| sign,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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