This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A196080 #20 Nov 14 2013 05:57:27
%S A196080 2,2,3,3,37,37,1111,1111,6913,6913,799933,799933,739138093,739138093,
%T A196080 44841044309,44841044309,32285551902481,32285551902481,
%U A196080 9879378882159187,9879378882159187,1251387991740163687
%N A196080 Numerators of the sum of the n-th partial sums of the expansions of e and 1/e.
%C A196080 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,.. .
%C A196080 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).)
%C A196080 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).
%e A196080 a(0)=1+1, a(1)=2+0, a(2)=(5+1)/2, a(3)=(8+1)/3.
%t A196080 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 *)
%Y A196080 Cf. A001113, A068985, A137204 (e+1/e).
%K A196080 nonn,frac
%O A196080 0,1
%A A196080 _Paul Curtz_, Sep 27 2011
%E A196080 Redefined by reduced fractions. - _R. J. Mathar_, Jul 02 2012