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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)!. 34

%I #14 May 14 2020 13:43:50

%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

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)