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A143413 Apéry-like numbers for the constant e: a(n) = 1/(n-1)!*Sum_{k = 0..n+1} (-1)^k*C(n+1,k)*(2*n-k)! for n >= 1.

%I A143413 #26 Jul 11 2021 02:56:04
%S A143413 -1,1,11,181,3539,81901,2203319,67741129,2346167879,90449857081,
%T A143413 3843107102339,178468044946621,8994348275804891,488964835817842021,
%U A143413 28523735794360301039,1777328098986754744081,117817961601577138782479,8279178465722546926265329
%N A143413 Apéry-like numbers for the constant e: a(n) = 1/(n-1)!*Sum_{k = 0..n+1} (-1)^k*C(n+1,k)*(2*n-k)! for n >= 1.
%C A143413 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 Napier's constant: e = 2 * Sum_{n >= 1} (-1)^n * n^2/(a(n)* a(n-1)).
%C A143413 Notice the striking parallels with 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)) = 5*[1/(1*3) + 1/(2^2*3*19) + 1/(3^2*19*147) + ...].
%H A143413 Seiichi Manyama, <a href="/A143413/b143413.txt">Table of n, a(n) for n = 0..365</a>
%H A143413 A. van der Poorten, <a href="http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf">A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report.</a>, Math. Intelligencer 1 (1978/79), no 4, 195-203.
%F A143413 a(0):= -1, a(n) = 1/(n-1)!*sum {k = 0..n+1} (-1)^k*C(n+1,k)*(2*n-k)! for n >= 1.
%F A143413 Apart from the initial term, this sequence is the second superdiagonal of the square array A060475; equivalently, the second subdiagonal of the square array A086764.
%F A143413 Recurrence relation: a(0) = -1, a(1) = 1, (n-1)^2*a(n) - n^2*a(n-2) = (2*n-1)*(2*n^2-2*n+1)*a(n-1), n >= 2.
%F A143413 Let b(n) denote the solution to this recurrence with initial conditions b(0) = 0, b(1) = 2. Then b(n) = A143414(n) = 1/(n-1)!*sum {k = 0..n-1} C(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) -> e very rapidly.
%F A143413 For example, b(100)/a(100) - e is approximately 1.934 * 10^(-436). The identity b(n)*a(n-1) - b(n-1)*a(n) = (-1)^n *2*n^2 leads to rapidly converging series for e and 1/e: e = 2 * Sum_{n >= 1} (-1)^n * n^2/(a(n)*a(n-1)) = 2*[1 + 2^2/(1*11) - 3^2/(11*181) + 4^2/(181*3539) - ...]; 1/e = 1/2 - 2*Sum_{n >= 2} (-1)^n * n^2/(b(n)*b(n-1)) = 1/2 - 2*[2^2/(2*30) - 3^2/(30*492) + 4^2/(492*9620) - ...].
%F A143413 Conjectural congruences: for r >= 0 and odd prime p, calculation suggests that a(p^r*(p+1)) + a(p^r) == 0 (mod p^(r+1)).
%F A143413 a(n) = ((2*n)!/(n-1)!)*hypergeom([-n-1], [-2*n], -1) for n >= 2. - _Peter Luschny_, Nov 14 2018
%F A143413 a(n) ~ 2^(2*n + 1/2) * n^(n+1) / exp(n + 1/2). - _Vaclav Kotesovec_, Jul 11 2021
%p A143413 a := n -> 1/(n-1)!*add((-1)^k*binomial(n+1,k)*(2*n-k)!, k = 0..n+1):
%p A143413 seq(a(n), n = 1..19);
%p A143413 # Alternative
%p A143413 a := n -> `if`(n<2, 2*n-1, (2*n)!/(n-1)!*hypergeom([-n-1], [-2*n], -1)):
%p A143413 seq(simplify(a(n)), n=0..17); # _Peter Luschny_, Nov 14 2018
%t A143413 Join[{-1}, Table[(1/(n-1)!)*Sum[(-1)^k*Binomial[n+1,k]*(2*n-k)!, {k, 0, n+1}], {n, 1, 50}]] (* _G. C. Greubel_, Oct 24 2017 *)
%o A143413 (PARI) concat([-1], for(n=1,25, print1((1/(n-1)!)*sum(k=0,n+1, (-1)^k*binomial(n+1,k)*(2*n-k)!), ", "))) \\ _G. C. Greubel_, Oct 24 2017
%Y A143413 Cf. A005258, A060475, A086764, A143414, A143415.
%Y A143413 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.)
%K A143413 easy,sign
%O A143413 0,3
%A A143413 _Peter Bala_, Aug 14 2008