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A143415 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)!.

%I A143415 #18 Oct 01 2024 23:20:08
%S A143415 0,1,5,41,481,7421,142601,3288205,88577021,2731868921,94969529101,
%T A143415 3675200329841,156725471006105,7302990263511541,369216917569411601,
%U A143415 20130327811188977621,1177435382675193700021,73546210385434763486705
%N A143415 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 A143415 This sequence is a modified version of A143414.
%H A143415 Seiichi Manyama, <a href="/A143415/b143415.txt">Table of n, a(n) for n = 0..366</a>
%F A143415 a(n) = 1/(n+1)!*sum {k = 0..n-1} C(n-1,k)*(2*n-k)!.
%F A143415 a(n) = 1/(n*(n+1))*A143414(n) for n > 0.
%F A143415 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 A143415 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 A143415 a(n) = ((2*n)!/(n+1)!)*hypergeom([1-n], [-2*n], 1) for n > 0. - _Peter Luschny_, May 14 2020
%p A143415 a := n -> 1/(n+1)!*add (binomial(n-1,k)*(2*n-k)!,k = 0..n-1): seq(a(n),n = 0..19);
%p A143415 # Alternative:
%p A143415 A143415 := n -> `if`(n=0, 0, ((2*n)!/(n+1)!)*hypergeom([1-n], [-2*n], 1)):
%p A143415 seq(simplify(A143415(n)), n = 0..17); # _Peter Luschny_, May 14 2020
%t A143415 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 A143415 (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 A143415 Cf. A143413, A143414.
%Y A143415 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 A143415 easy,nonn
%O A143415 0,3
%A A143415 _Peter Bala_, Aug 14 2008