The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A000522

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A000522

Total number of ordered k-tuples (k=0..n) of distinct elements from an n-element set: a(n) = Sum_{k=0..n} n!/k!.
(history; published version)

#501 by Alois P. Heinz at Thu Oct 31 13:48:20 EDT 2024

STATUS

proposed

approved

#500 by Michel Marcus at Sat Oct 26 14:16:56 EDT 2024

STATUS

editing

proposed

Discussion

Thu Oct 31

13:47

Alois P. Heinz: first read, then write ...

#499 by Michel Marcus at Sat Oct 26 14:16:44 EDT 2024

COMMENTS

a(n)/n! = e as a(n) increases to inf. Adam Sniffen, Oct 26 2024

LINKS

J. Jonathan Sondow, <a href="https://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641. arXiv:0704.1282 [math.HO], 2007-2010.

J. Jonathan Sondow, <a href="https://home.earthlink.net/~jsondow/PrimesAndE.pdf">The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection</a>

J. Jonathan Sondow and K. Schalm, <a href="https://arxiv.org/abs/0709.0671">Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II</a>, arXiv:0709.0671 [math.NT], 2006-2009; Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

FORMULA

Limit_{n->infinityoo} a(n)/n! = e = 1/(1-1/(2-1/(3-2/(4-...-n/((n+2)-...))))). This is the particular case m = 0 of the general result m!/e - d_m = (-1)^(m+1) *(1/(m+2 -1/(m+3 -2/(m+4 -3/(m+5 -...))))), where d_m denotes the m-th derangement number A000166(m).

STATUS

proposed

editing

Discussion

Sat Oct 26

14:16

Michel Marcus: so removed

#498 by Adam Sniffen at Sat Oct 26 12:47:58 EDT 2024

STATUS

editing

proposed

Discussion

Sat Oct 26

12:58

Amiram Eldar: This already exists. See Peter Bala's Jul 15 2008 formula: Limit_{n->infinity} a(n)/n! = e.

#497 by Adam Sniffen at Sat Oct 26 12:46:38 EDT 2024

COMMENTS

a(n)/n! = e as a(n) increases to inf. Adam Sniffen, Oct 26 2024

STATUS

approved

editing

#496 by Michael De Vlieger at Sun Oct 13 07:09:21 EDT 2024

STATUS

reviewed

approved

#495 by Joerg Arndt at Sun Oct 13 01:36:43 EDT 2024

STATUS

proposed

reviewed

#494 by Stefano Spezia at Sat Oct 12 17:01:35 EDT 2024

STATUS

editing

proposed

#493 by Stefano Spezia at Sat Oct 12 15:56:09 EDT 2024

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E11.

STATUS

approved

editing

#492 by Michael De Vlieger at Mon May 20 11:55:31 EDT 2024

STATUS

reviewed

approved