login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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.

MAPLE

with(combinat): a := n -> 1/(n+1)!*add (binomial(n-1, k)*(2*n-k)!, k = 0..n-1): seq(a(n), n = 0..19);

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: A049119 A032188 A240996 * A056545 A275787 A009755

Adjacent sequences:  A143412 A143413 A143414 * A143416 A143417 A143418

KEYWORD

easy,nonn

AUTHOR

Peter Bala, Aug 14 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 20 15:54 EST 2018. Contains 299380 sequences. (Running on oeis4.)