%I
%S 1,1,0,2,8,0,320,2800,0,344960,4659200,0,1172864000,21423001600,0,
%T 9117844736000,209945415680000,0,135381758640128000,
%U 3761801958154240000,0,3421097040836034560000,111349337961365504000000,0,135776499356700539617280000
%N Exponential expansion of the real root y = y(x) of y^3  3*x*y  1.
%C This is an example of an application of Ramanujan's Master theorem for definite integrals; see eq. (B) on p. 186 of the Hardy reference. This application is given under (ii) on pp. 194195; here with r = 1, p = 1, q = 2, and x and a there are y and x here, respectively.
%C The general formula for the exponential expansion of the rth power of the solution y=y(x) of y^q  q*x*y  1 = 0 which starts with y(0) = 1 is y(x)^r = Sum_{n>=0} lambda(n;r,q,p)*x^n/n! with lambda(0;r,q,p) = 1, lambda(1;r,q,p) = r and lambda(n;r,q,p) = r*Product_{j=1..n1} (r + n*p  q*j) for n >= 2. Hardy gives a convergence condition for theorem (B) on p. 189: the class K(A,P,delta) for phi(u) = lambda(u) / Gamma(1+u), u complex, here for lambda(u) = lambda(u;r,q,p).
%D G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, ch, XI, pp. 186211.
%H G. C. Greubel, <a href="/A282626/b282626.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) = Product_{j=1..n1} (n + 1  3*j), n >=0 (empty product = 1).
%F E.g.f.: ((1 + sqrt(14*x^3))/2)^(1/3) + x/((1 + sqrt(14*x^3))/2)^(1/3).
%F E.g.f.: ((1 + sqrt(14*x^3))/2)^(1/3) + ((1  sqrt(14*x^3))/2)^(1/3).
%t Table[Product[n+13*j, {j,1,n1}], {n,0,25}] (* _G. C. Greubel_, Mar 29 2019 *)
%o (PARI) vector(25, n, n; prod(j=1,n1, (n+13*j))) \\ _G. C. Greubel_, Mar 29 2019
%o (MAGMA) [1,1] cat [(&*[n+13*j: j in [1..(n1)]]): n in [2..25]]; // _G. C. Greubel_, Mar 29 2019
%o (Sage) [1] + [product(n+13*j for j in (1..(n1))) for n in (1..25)] # _G. C. Greubel_, Mar 29 2019
%Y Cf. A282627.
%K sign,easy
%O 0,4
%A _Wolfdieter Lang_, Mar 04 2017
%E More terms from _G. C. Greubel_, Mar 29 2019
