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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
0, 1, 5, 41, 481, 7421, 142601, 3288205, 88577021, 2731868921, 94969529101, 3675200329841, 156725471006105, 7302990263511541, 369216917569411601, 20130327811188977621, 1177435382675193700021, 73546210385434763486705 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This sequence is a modified version of A143414.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..366

FORMULA

a(n) = 1/(n+1)!*sum {k = 0..n-1} C(n-1,k)*(2*n-k)!.

a(n) = 1/(n*(n+1))*A143414(n) for n > 0.

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) + ...] .

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.

a(n) = ((2*n)!/(n+1)!)*hypergeom([1-n], [-2*n], 1)) for n > 0. - Peter Luschny, May 14 2020

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:

A143415 := n -> `if`(n=0, 0, ((2*n)!/(n+1)!)*hypergeom([1-n], [-2*n], 1)):

seq(simplify(A143415(n)), n = 0..17); # 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

Cf. A143413, A143414.

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.)

Sequence in context: A305981 A032188 A240996 * A056545 A325888 A275787

Adjacent sequences:  A143412 A143413 A143414 * A143416 A143417 A143418

KEYWORD

easy,nonn

AUTHOR

Peter Bala, Aug 14 2008

STATUS

approved

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Last modified October 20 19:28 EDT 2020. Contains 337909 sequences. (Running on oeis4.)