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%I M0415 #85 Sep 23 2022 16:17:45
%S 1,-1,1,0,-1,1,-1,1,0,-1,2,-3,2,0,-2,4,-4,3,-1,-3,6,-7,5,0,-5,9,-10,7,
%T -1,-7,14,-16,11,-1,-11,20,-22,16,-2,-15,29,-33,23,-2,-23,41,-45,32,
%U -4,-30,57,-64,45,-4,-43,78,-86,60,-7,-57,107,-119,83,-8,-79,143
%N G.f.: Product_{k>0} (1-x^(5k-1))*(1-x^(5k-4))/((1-x^(5k-2))*(1-x^(5k-3))).
%C Expansion of f(-x, -x^4) / f(-x^2, -x^3) in powers of x where f(,) is Ramanujan's two-variable theta function.
%C Hauptmodul series for Gamma(5).
%C Expansion of Rogers-Ramanujan's continued fraction 1 / (1 + x / ( 1 + x^2 / ( 1 + x^3 / ( 1 + x^4 / ... )))).
%C Given the g.f. A(x) the notation R(q) := q^(1/5) * A(q) is used by Berndt.
%D G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005, see p. 57.
%D B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 9.
%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 81.
%D A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 24.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A007325/b007325.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H G. E. Andrews, <a href="http://www.jstor.org/stable/2974472">Simplicity and surprise in Ramanujan's "Lost" Notebook</a>, Amer. Math. Monthly, 104 (No. 10, Dec. 1997), 918-925.
%H W. Duke, <a href="http://dx.doi.org/10.1090/S0273-0979-05-01047-5">Continued fractions and modular functions</a>, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eq. (6.4).
%H G. S. Joyce, <a href="http://dx.doi.org/10.1088/0305-4470/21/20/005">Exact results for the activity and thermal compressibility of the hard-hexagon model</a>, J. Phys. A 21 (1988), L983-L988.
%H J. Malenfant, <a href="http://arxiv.org/abs/1109.5957">Generalizing Ramanujan's J Functions</a>, arXiv preprint arXiv:1109.5957 [math.NT], 2011.
%H P. J. Nahin, <a href="http://press.princeton.edu/titles/9530.html">Number-Crunching</a>, Princeton University Press, 2011. See p. 22 equation (2.2.4).
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction.</a>
%F Euler transform of period 5 sequence [-1, 1, 1, -1, 0, ...] (=-A080891).
%F G.f.: Product_{k>=1}(1-x^(5*k-1)) * (1-x^(5*k-4)) / ( (1-x^(5*k-2)) * (1-x^(5*k-3)) ) = H(x) / G(x) where H and G are respectively the g.f. of A003114 and A003106.
%F G.f.: (Sum (-1)^k x^((5*k + 3)*k/2))/(Sum (-1)^k x^((5*k + 1)*k/2)). - _Michael Somos_, Dec 13 2002
%F Given g.f. A(x), then B(q) = q * A(q^5) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v + u*v^3 + u^3*v^2. - _Michael Somos_, Mar 09 2004
%F Given g.f. A(x), then B(q) = q * A(q^5) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u * (u*v + w^2 + v^2*w) - w. - _Michael Somos_, Aug 29 2005
%F Given g.f. A(x), then B(q) = q * A(q^5) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u2 + u1*u3^2*u6 + u2*u3^2 - u2^2*u3*u6 - u3. - _Michael Somos_, Aug 29 2005
%F G.f.: 1 / (1 + x / ( 1 + x^2 / ( 1 + x^3 / ( 1 + x^4 / ... )))).
%F G.f.: 1 / (1 + 1 / (x^-1 + 1 / (x^-1 + 1 / (x^-2 + 1 / (x^-2 + 1 / ... ))))). - _Michael Somos_, Apr 30 2012
%F G.f.: A(x) = S(0) -1; S(k) = 1 + x^k/S(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Dec 18 2011
%F Hankel transform is A167683. - _Michael Somos_, Apr 30 2012
%F a(n) = (-1)^n * A226556(n). - _Michael Somos_, Jun 11 2013
%F a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A109091(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 01 2017
%e G.f. = 1 - x + x^2 - x^4 + x^5 - x^6 + x^7 - x^9 + 2*x^10 - 3*x^11 + 2*x^12 - ...
%e G.f. = q - q^6 + q^11 - q^21 + q^26 - q^31 + q^36 - q^46 + 2*q^51 - 3*q^56 + ...
%p t1:=mul((1-x^(5*k-1))*(1-x^(5*k-4))/((1-x^(5*k-2))*(1-x^(5*k-3))), k=1..60); seriestolist(series(t1,x,59)); # _N. J. A. Sloane_, Jun 10 2013
%p A007325_G:=proc(x,NK);Digits:=250;
%p Q2:=1;
%p for k from NK by -1 to 0 do
%p Q1:=1+x^k/Q2; Q2:=Q1; od;
%p Q3:=Q2; S:=Q3-1;
%p end;
%p # _Sergei N. Gladkovskii_, Dec 18 2011
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^5] QPochhammer[ q^4, q^5] / (QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5]), {q, 0, n}]; (* _Michael Somos_, Aug 17 2011 *)
%t a[ n_] := SeriesCoefficient[ ContinuedFractionK[ q^k, 1, {k, 0, n}], {q, 0, n}]; (* _Michael Somos_, Jun 10 2013 *)
%t max = 65; CoefficientList[ Series[ Fold[ #2/(1 + #1)&, q^n, q^Reverse[ Range[0, max-1] ] ], {q, 0, max}], q] (* _Jean-François Alcover_, Apr 04 2013 *)
%o (PARI) {a(n) = my(k); if( n<0, 0, k = (3 + sqrtint(9 + 40*n)) \ 10; polcoeff( sum( n=-k, k, (-1)^n * x^((5*n^2 + 3*n)/2), x * O(x^n)) / sum( n=-k, k,( -1)^n * x^((5*n^2 + n)/2), x * O(x^n)), n))};
%o (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))};
%o (PARI) {a(n) = my(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[2, 1] / cf[1, 1] + x * O(x^n), n))};
%o (PARI) {a(n) = my(A, m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=5; A = x * subst(A, x, x^5); A = (A * (1 - 2*A + 4*A^2 - 3*A^3 + A^4) / (1 + 3*A + 4*A^2 + 2*A^3 + A^4) / x)^(1/5)); polcoeff(A, n))};
%Y Cf. A055101, A055102, A055103, A003823, A167683.
%K sign,easy,nice
%O 0,11
%A _N. J. A. Sloane_, _Mira Bernstein_
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