A120266 Numerator of Sum_{k=0..n} n^k/k!.

2, 5, 13, 103, 1097, 1223, 47273, 556403, 10661993, 7281587, 62929017101, 7218065, 60718862681977, 595953719897, 13324966405463, 247016301114823, 28505097599389815853, 549689343118061, 320305944459287485595917

Offset: 1

Author

Alexander Adamchuk, Jun 30 2006

Keywords

• frac
• nonn

Comments

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/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

Examples

			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
		

Links

• Eric Weisstein, Exponential Sum Function.

Crossrefs

Denominators are A214401. Cf. also A063170, A090878, A119029, A120267, A214402.

Programs

Mathematica

Numerator[Table[Sum[n^k/k!, {k,0,n}], {n,1,30}]]

Formula

a(n) = numerator(Sum_{k=0..n} n^k/k!).

a(n) = A063170(n)/A214402(n) = (n!/A214402(n))*Sum_{k=0..n} n^k/k! for n > 0. - Jonathan Sondow, Jul 16 2012