This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A282626 #21 Sep 08 2022 08:46:18
%S A282626 1,1,0,-2,8,0,-320,2800,0,-344960,4659200,0,-1172864000,21423001600,0,
%T A282626 -9117844736000,209945415680000,0,-135381758640128000,
%U A282626 3761801958154240000,0,-3421097040836034560000,111349337961365504000000,0,-135776499356700539617280000
%N A282626 Exponential expansion of the real root y = y(x) of y^3 - 3*x*y - 1.
%C A282626 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 A282626 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 A282626 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 A282626 G. C. Greubel, <a href="/A282626/b282626.txt">Table of n, a(n) for n = 0..400</a>
%F A282626 a(n) = Product_{j=1..n-1} (n + 1 - 3*j), n >=0 (empty product = 1).
%F A282626 E.g.f.: ((1 + sqrt(1-4*x^3))/2)^(1/3) + x/((1 + sqrt(1-4*x^3))/2)^(1/3).
%F A282626 E.g.f.: ((1 + sqrt(1-4*x^3))/2)^(1/3) + ((1 - sqrt(1-4*x^3))/2)^(1/3).
%t A282626 Table[Product[n+1-3*j, {j,1,n-1}], {n,0,25}] (* _G. C. Greubel_, Mar 29 2019 *)
%o A282626 (PARI) vector(25, n, n--; prod(j=1,n-1, (n+1-3*j))) \\ _G. C. Greubel_, Mar 29 2019
%o A282626 (Magma) [1,1] cat [(&*[n+1-3*j: j in [1..(n-1)]]): n in [2..25]]; // _G. C. Greubel_, Mar 29 2019
%o A282626 (Sage) [1] + [product(n+1-3*j for j in (1..(n-1))) for n in (1..25)] # _G. C. Greubel_, Mar 29 2019
%Y A282626 Cf. A282627.
%K A282626 sign,easy
%O A282626 0,4
%A A282626 _Wolfdieter Lang_, Mar 04 2017
%E A282626 More terms from _G. C. Greubel_, Mar 29 2019