cp's OEIS Frontend

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

%I A195326 #38 May 05 2019 03:48:46
%S A195326 0,2,2,7,7,47,47,5923,5923,426457,426457,15636757,15636757,7318002277,
%T A195326 7318002277,1536780478171,1536780478171,603180793741,603180793741,
%U A195326 142957467201379447,142957467201379447
%N A195326 Numerators of fractions leading to e - 1/e (A174548).
%C A195326 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, ...
%C A195326 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.
%C A195326 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.
%C A195326 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).
%e A195326 a(0) =  1  -  1;
%e A195326 a(1) =  2  -  0;
%e A195326 a(2) = 5/2 - 1/2.
%p A195326 taylExp1 := proc(n)
%p A195326         add(1/j!,j=0..n) ;
%p A195326 end proc:
%p A195326 A000255 := proc(n)
%p A195326         if n <=1 then
%p A195326                 1;
%p A195326         else
%p A195326                 n*procname(n-1)+(n-1)*procname(n-2) ;
%p A195326         end if;
%p A195326 end proc:
%p A195326 A001048 := proc(n)
%p A195326         n!+(n-1)! ;
%p A195326 end proc:
%p A195326 A195326 := proc(n)
%p A195326         if n = 0 then
%p A195326                 0;
%p A195326         elif n =1 then
%p A195326                 2;
%p A195326         else
%p A195326                 taylExp1(n) -A000255(n-2)/A001048(n-1);
%p A195326         end if;
%p A195326           numer(%);
%p A195326 end proc:
%p A195326 seq(A195326(n),n=0..20) ; # _R. J. Mathar_, Oct 14 2011
%Y A195326 Cf. A001113, A068985, A197222, A197223.
%K A195326 nonn,frac
%O A195326 0,2
%A A195326 _Paul Curtz_, Oct 12 2011
%E A195326 Material meant to be placed in other sequences removed by _R. J. Mathar_, Oct 14 2011