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A195326 Numerators of fractions leading to e - 1/e (A174548). 1
0, 2, 2, 7, 7, 47, 47, 5923, 5923, 426457, 426457, 15636757, 15636757, 7318002277, 7318002277, 1536780478171, 1536780478171, 603180793741, 603180793741, 142957467201379447, 142957467201379447 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The sequence of approximations of exp(1) obtained by truncating the Taylor series of exp(x) after n terms is A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, ...
A Taylor series of exp(-1) is 1, 0, 1/2, 1/3, 3/8, ... and (apart from the first 2 terms) given by A000255(n)/A001048(n). Subtracting both sequences term by term we obtain a series for exp(1) - exp(-1) = 0, 2, 2, 7/3, 7/3, 47/20, 47/20, 5923/2520, 5923/2520, 426457/181440, 426457/181440, ... which defines the numerators here.
Each second of the denominators (that is 3, 2520, 19958400, ...) is found in A085990 (where each third term, that is 60, 19958400, ...) is to be omitted.
This numerator sequence here is basically obtained by doubling entries of A051397, A009628, A087208, or A186763, caused by the standard associations between cosh(x), sinh(x) and exp(x).
LINKS
EXAMPLE
a(0) = 1 - 1;
a(1) = 2 - 0;
a(2) = 5/2 - 1/2.
MAPLE
taylExp1 := proc(n)
add(1/j!, j=0..n) ;
end proc:
A000255 := proc(n)
if n <=1 then
1;
else
n*procname(n-1)+(n-1)*procname(n-2) ;
end if;
end proc:
A001048 := proc(n)
n!+(n-1)! ;
end proc:
A195326 := proc(n)
if n = 0 then
0;
elif n =1 then
2;
else
taylExp1(n) -A000255(n-2)/A001048(n-1);
end if;
numer(%);
end proc:
seq(A195326(n), n=0..20) ; # R. J. Mathar, Oct 14 2011
CROSSREFS
Sequence in context: A199886 A117779 A300952 * A263868 A263793 A232647
KEYWORD
nonn,frac
AUTHOR
Paul Curtz, Oct 12 2011
EXTENSIONS
Material meant to be placed in other sequences removed by R. J. Mathar, Oct 14 2011
STATUS
approved

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Last modified April 15 14:58 EDT 2024. Contains 371692 sequences. (Running on oeis4.)