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

Author

Wolfdieter Lang, Mar 04 2017

Keywords

• sign
• easy

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

Crossrefs

Cf. A282626.

Programs

Magma

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

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 *)

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

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

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).

Extensions

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