A282627 Exponential expansion of the square of the real root y = y(x) of y^3 - 3*x*y - 1.

1, 2, 2, -4, 0, 80, -560, 0, 49280, -582400, 0, 117286400, -1947545600, 0, 701372672000, -14996101120000, 0, 8461359915008000, -221282468126720000, 0, 180057738991370240000, -5567466898068275200000, 0, 6171659061668206346240000, -220248990487580966912000000, 0

Offset

0,2

Comments

See a comment on A282626 on Ramanujan's Master theorem (B) for definite integral, and the Hardy reference.

This is the exponential (aka binomial) convolution of A282626.

Links

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

Formula

a(n) = 2*Product_{j=1..n-1}(n + 2 - 3*j), n >= 1, (empty product = 1) and a(0) = 1.

E.g.f.: Square of the e.g.f. of A282626 =

2*x + ((1 + sqrt(1-4*x^3))/2)^(2/3) + ((1 - sqrt(1-4*x^3))/2)^(2/3).

Mathematica

Table[If[n==0, 1, 2*Product[n+2-3*j, {j, 1, n-1}]], {n, 0, 25}] (* G. C. Greubel, Mar 29 2019 *)

Prog

(PARI) vector(25, n, n--; if(n==0, 1, 2*prod(j=1, n-1, (n+2-3*j)))) \\ G. C. Greubel, Mar 29 2019
(Magma) [1, 2] cat [2*(&*[n+2-3*j: j in [1..(n-1)]]): n in [2..25]]; // G. C. Greubel, Mar 29 2019
(Sage) [1] + [2*product(n+2-3*j for j in (1..(n-1))) for n in (1..25)] # G. C. Greubel, Mar 29 2019

Crossrefs

Cf. A282626.

Sequence in context: A240491 A113750 A355204 * A004565 A068449 A362258

Adjacent sequences: A282624 A282625 A282626 * A282628 A282629 A282630

Keyword

sign,easy

Author

Wolfdieter Lang, Mar 04 2017

Extensions

Terms a(21) onward added by G. C. Greubel, Mar 30 2019

Status

approved