|
| |
|
|
A196080
|
|
Numerators of the sum of the n-th partial sums of the expansions of e and 1/e.
|
|
0
|
|
|
|
2, 2, 3, 3, 37, 37, 1111, 1111, 6913, 6913, 799933, 799933, 739138093, 739138093, 44841044309, 44841044309, 32285551902481, 32285551902481, 9879378882159187, 9879378882159187, 1251387991740163687
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
COMMENTS
|
The n-th partial sums of the Taylor expansion of E are f(n) = A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, 163/60,.. .
The partial sums of an expansion of 1/e are essentially A000255(n-2)/A001048(n-1) preceded by 1 and 0, namely g(n)= 1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760,... (Jolley's partial sums of 1/E in A068985 is the bisection 0, 1/3, 11/30, 103/280, 16687/45360,... of g(n).)
The current sequence are the numerators of f(n)+g(n), converging to E+1/E, namely 2, 2, 3, 3, 37/12, 37/12, 1111/360, 1111/360, 6913/2240 = 62217/21060, 6913/2240 = 62217/21060, 799933/259200 = 5599531/1814400,... The unreduced fractions are apparently given by duplicated A051396(n+1)/A002674(n).
|
|
|
LINKS
|
Table of n, a(n) for n=0..20.
|
|
|
EXAMPLE
|
a(0)=1+1, a(1)=2+0, a(2)=(5+1)/2, a(3)=(8+1)/3.
|
|
|
MATHEMATICA
|
a[n_] := (E*Gamma[n+1, 1] + (1/E)*Gamma[n+1, -1])/n! // FullSimplify // Numerator; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 02 2012 *)
|
|
|
CROSSREFS
|
Cf. A001113, A068985, A137204 (e+1/e).
Sequence in context: A180000 A053094 A309283 * A124516 A205958 A309090
Adjacent sequences: A196077 A196078 A196079 * A196081 A196082 A196083
|
|
|
KEYWORD
|
nonn,frac
|
|
|
AUTHOR
|
Paul Curtz, Sep 27 2011
|
|
|
EXTENSIONS
|
Redefined by reduced fractions. - R. J. Mathar, Jul 02 2012
|
|
|
STATUS
|
approved
|
| |
|
|
