Template:Sequence of the Day for May 19

Intended for: May 19, 2014

Timetable

• First draft entered by Alonso del Arte on May 9, 2013 ✓
• Draft reviewed by Daniel Forgues on May 18, 2015 (May 15, 2016) ✓
• Draft to be approved by April 19, 2014

Yesterday's SOTD * Tomorrow's SOTD

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A001142:

∏ n k  = 1  k  2 k   − 1 −  n

, or

∏ n k  = 0  (  nk  )  , n   ≥   0

.

{ 1, 1, 2, 9, 96, 2500, 162000, ... }

The second formula above shows that this sequence is also the products of consecutive horizontal rows of Pascal’s triangle. As it happens, these numbers are related to Euler’s number

e

thus:
     

n  →  ∞limn  →  ∞  ⎛ ⎝    a (n + 1) a (n)    ⎞ ⎠ ⎛ ⎝    a (n) a (n − 1)    ⎞ ⎠ = n  →  ∞limn  →  ∞  a (n − 1) a (n + 1) a (n) 2 = e.