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 A282626 Exponential expansion of the real root y = y(x) of y^3 - 3*x*y - 1. 2

%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. 194-195; 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 r-th 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..n-1} (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. 186-211.

%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..n-1} (n + 1 - 3*j), n >=0 (empty product = 1).

%F E.g.f.: ((1 + sqrt(1-4*x^3))/2)^(1/3) + x/((1 + sqrt(1-4*x^3))/2)^(1/3).

%F E.g.f.: ((1 + sqrt(1-4*x^3))/2)^(1/3) + ((1 - sqrt(1-4*x^3))/2)^(1/3).

%t Table[Product[n+1-3*j, {j,1,n-1}], {n,0,25}] (* _G. C. Greubel_, Mar 29 2019 *)

%o (PARI) vector(25, n, n--; prod(j=1,n-1, (n+1-3*j))) \\ _G. C. Greubel_, Mar 29 2019

%o (MAGMA) [1,1] cat [(&*[n+1-3*j: j in [1..(n-1)]]): n in [2..25]]; // _G. C. Greubel_, Mar 29 2019

%o (Sage)  + [product(n+1-3*j for j in (1..(n-1))) 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

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Last modified August 16 05:55 EDT 2022. Contains 356160 sequences. (Running on oeis4.)