%I
%S 2,5,13,103,1097,1223,47273,556403,10661993,7281587,62929017101,
%T 7218065,60718862681977,595953719897,13324966405463,247016301114823,
%U 28505097599389815853,549689343118061,320305944459287485595917
%N Numerator of Sum_{k=0..n} n^k/k!.
%C Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious.  _Petros Hadjicostas_, May 12 2020
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/ExponentialSumFunction.html">Exponential Sum Function</a>.
%F a(n) = numerator(Sum_{k=0..n} n^k/k!).
%F a(n) = A063170(n)/A214402(n) = (n!/A214402(n))*Sum_{k=0..n} n^k/k! for n > 0.  _Jonathan Sondow_, Jul 16 2012
%e The first few fractions are 2, 5, 13, 103/3, 1097/12, 1223/5, 47273/72, 556403/315, 10661993/2240, ... = A120266/A214401.  _Petros Hadjicostas_, May 12 2020
%t Numerator[Table[Sum[n^k/k!, {k,0,n}], {n,1,30}]]
%Y Denominators are A214401. Cf. also A063170, A090878, A119029, A120267, A214402.
%K frac,nonn
%O 1,1
%A _Alexander Adamchuk_, Jun 30 2006
