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

1, 1, 0, -2, 8, 0, -320, 2800, 0, -344960, 4659200, 0, -1172864000, 21423001600, 0, -9117844736000, 209945415680000, 0, -135381758640128000, 3761801958154240000, 0, -3421097040836034560000, 111349337961365504000000, 0, -135776499356700539617280000

Offset: 0

Author

Wolfdieter Lang, Mar 04 2017

Keywords

• sign
• easy

Comments

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.

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).

References

• 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A282627.

Programs

Magma

[1,1] cat [(&*[n+1-3*j: j in [1..(n-1)]]): n in [2..25]]; // G. C. Greubel, Mar 29 2019

Mathematica

Table[Product[n+1-3*j, {j,1,n-1}], {n,0,25}] (* G. C. Greubel, Mar 29 2019 *)

PARI

vector(25, n, n--; prod(j=1,n-1, (n+1-3*j))) \\ G. C. Greubel, Mar 29 2019

Sage

[1] + [product(n+1-3*j for j in (1..(n-1))) for n in (1..25)] # G. C. Greubel, Mar 29 2019

Formula

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

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

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

Extensions

More terms from G. C. Greubel, Mar 29 2019