%I #18 Oct 01 2024 23:20:08
%S 0,1,5,41,481,7421,142601,3288205,88577021,2731868921,94969529101,
%T 3675200329841,156725471006105,7302990263511541,369216917569411601,
%U 20130327811188977621,1177435382675193700021,73546210385434763486705
%N Another sequence of Apery-like numbers for the constant 1/e: a(n) = 1/(n+1)!*Sum_{k = 0..n-1} C(n-1,k)*(2*n-k)!.
%C This sequence is a modified version of A143414.
%H Seiichi Manyama, <a href="/A143415/b143415.txt">Table of n, a(n) for n = 0..366</a>
%F a(n) = 1/(n+1)!*sum {k = 0..n-1} C(n-1,k)*(2*n-k)!.
%F a(n) = 1/(n*(n+1))*A143414(n) for n > 0.
%F Recurrence relation: a(0) = 0, a(1) = 1, (n-1)*(n+1)*a(n) - (n-2)*n*a(n-2) = (2*n-1)*(2*n^2-2*n+1)*a(n-1) for n >= 2. 1/e = 1/2 - 2 * Sum_{n = 1..inf} (-1)^(n+1)/(n*(n+2)*a(n)*a(n+1)) = 1/2 - 2*[1/(3*1*5) - 1/(8*5*41) + 1/(15*41*481) - 1/(24*481*7421) + ...] .
%F Conjectural congruences: for r >= 0 and prime p, calculation suggests the congruences a(p^r*(p+1)) == a(p^r) (mod p^(r+1)) may hold.
%F a(n) = ((2*n)!/(n+1)!)*hypergeom([1-n], [-2*n], 1) for n > 0. - _Peter Luschny_, May 14 2020
%p a := n -> 1/(n+1)!*add (binomial(n-1,k)*(2*n-k)!,k = 0..n-1): seq(a(n),n = 0..19);
%p # Alternative:
%p A143415 := n -> `if`(n=0, 0, ((2*n)!/(n+1)!)*hypergeom([1-n], [-2*n], 1)):
%p seq(simplify(A143415(n)), n = 0..17); # _Peter Luschny_, May 14 2020
%t 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 *)
%o (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
%Y Cf. A143413, A143414.
%Y 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 easy,nonn
%O 0,3
%A _Peter Bala_, Aug 14 2008