OFFSET
0,2
COMMENTS
This sequence satisfies the recursion (n-1)^2*a(n) - n^2*a(n-2) = (2*n-1)*(2*n^2-2*n+1)*a(n-1), which leads to a rapidly converging series for the constant 1/e: 1/e = 1/2 - 2 * Sum_{n >= 2} (-1)^n * n^2/(a(n)*a(n-1)).
Notice the striking resemblance to the theory of the Apéry numbers A(n) = A005258(n), which satisfy a similar recurrence relation n^2*A(n) - (n-1)^2*A(n-2) = (11*n^2-11*n+3)*A(n-1) and which appear in the series acceleration formula zeta(2) = 5*Sum_{n>=1} 1/(n^2*A(n)*A(n-1)). Compare with A143413 and A143415.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..365
A. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report, Math. Intelligencer 1 (1978/79), no. 4, 195-203.
FORMULA
a(n) = (1/(n-1)!)*Sum_{k = 0..n-1} binomial(n-1,k)*(2*n-k)!.
Recurrence relation: a(0) = 0, a(1) = 2, (n-1)^2*a(n) - n^2*a(n-2) = (2*n-1)*(2*n^2-2*n+1)*a(n-1), n >= 2.
Let b(n) denote the solution to this recurrence with initial conditions b(0) = -1, b(1) = 1. Then b(n) = A143413(n) = (1/(n-1)!)*Sum_{k = 0..n+1} (-1)^k*binomial(n+1,k)*(2*n-k)!.
The rational number b(n)/a(n) is equal to the Padé approximation to exp(x) of degree (n+1,n-1) evaluated at x = -1 and b(n)/a(n) -> 1/e very rapidly. For example, |b(100)/a(100) - 1/e| is approximately 2.177 * 10^(-437).
The identity a(n)*b(n-1) - a(n-1)*b(n) = (-1)^n *2*n^2 leads to rapidly converging series for the constants 1/e and e: 1/e = 1/2 - 2*Sum_{n >= 2} (-1)^n * n^2/(a(n)*a(n-1)) = 1/2 - 2*(2^2/(2*30) - 3^2/(30*492) + 4^2/(492*9620) - ...); e = 2 * Sum_{n >= 1} (-1)^n * n^2/(b(n)*b(n-1)) = 2*(1 + 2^2/(1*11) - 3^2/(11*181) + 4^2/(181*3539) - ...).
a(n) = (BesselK(n-1/2,1/2)-(1-2*n)*BesselK(n+1/2,1/2)) * exp(1/2)/(2*Pi^(1/2)). - Mark van Hoeij, Nov 12 2009
a(n) = ((2*n)!/(n-1)!)*hypergeom([1-n], [-2*n], 1) for n > 0. - Peter Luschny, May 14 2020
a(n) ~ 2^(2*n + 1/2) * n^(n+1) / exp(n - 1/2). - Vaclav Kotesovec, Jul 11 2021
MAPLE
a := n -> 1/(n-1)!*add (binomial(n-1, k)*(2*n-k)!, k = 0..n-1): seq(a(n), n = 0..19);
# Alternative:
A143414 := n -> `if`(n=0, 0, ((2*n)!/(n-1)!)*hypergeom([1-n], [-2*n], 1)):
seq(simplify(A143414(n)), n = 0..16); # Peter Luschny, May 14 2020
MATHEMATICA
Table[(1/(n-1)!)*Sum[Binomial[n-1, k]*(2*n-k)!, {k, 0, n-1}], {n, 0, 50}] (* G. C. Greubel, Oct 24 2017 *)
PROG
(PARI) for(n=0, 25, print1((1/(n-1)!)*sum(k=0, n-1, binomial(n-1, k)*(2*n-k)!), ", ")) \\ G. C. Greubel, Oct 24 2017
CROSSREFS
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
KEYWORD
easy,nonn
AUTHOR
Peter Bala, Aug 14 2008
STATUS
approved