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$$a_0 = 1, \ \ a_{n+1} = \ 1+\frac{n *a_n}{n+a_n} , \ \ n=0,1,2,3,4,...$$

I have built the above recursive formula. Some terms of sequence are: 1, 1, 3/2, 13/7, 73/34, 501/209, 4051/1546, 37633/13327, 394353/130922, 4596553/1441729, 58941091/17572114, 824073141/234662231,... Τhe numerators of the fractions are identical to the sequence A000262 in OEIS encyclopedia and the denominators to the A002720. If we take n = {inf,.......,5,4,3,2,1}, ie to start from quite high and finish at 1. Let a (about 1.6768...) the last term. Then 1/(a*exp(1)) converges to: 0.21938393439552027367716377546012164903 ... that is the decimal expansion of -Ei(-1), A099285 in OEIS.

My question is: How it is explained, and how does the formula related to the above sequences?

$$a_0 = 1, \ \ a_{n+1} = \ 1+\frac{n *a_n}{n+a_n} , \ \ n=0,1,2,3,4,...$$

I have built the above recursive formula. Some terms of sequence are: 1, 1, 3/2, 13/7, 73/34, 501/209, 4051/1546, 37633/13327, 394353/130922, 4596553/1441729, 58941091/17572114, 824073141/234662231,... Τhe numerators of the fractions are identical to the sequence A000262 in OEIS encyclopedia and the denominators to the A002720. If we take n = {inf,.......,5,4,3,2,1}, ie to start from quite high and finish at 1. Let a (about 1.6768...) the last term. Then 1/(a*exp(1)) converges to: 0.21938393439552027367716377546012164903 ... that is the decimal expansion of -Ei(-1), A099285 in OEIS.

My question is: How it is explained, and how does the formula relate to the above sequences?