A195326 Numerators of fractions leading to e - 1/e (A174548).

0, 2, 2, 7, 7, 47, 47, 5923, 5923, 426457, 426457, 15636757, 15636757, 7318002277, 7318002277, 1536780478171, 1536780478171, 603180793741, 603180793741, 142957467201379447, 142957467201379447

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

Table of n, a(n) for n=0..20.

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

Cf. A001113, A068985, A197222, A197223.

Sequence in context: A376814 A117779 A300952 * A263868 A263793 A232647

Adjacent sequences: A195323 A195324 A195325 * A195327 A195328 A195329

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