Does it exist a recursive formula that approximates $\sum_{k=1}^n k^k\,$?

Question

Does it exist a recursive formula that approximates $\sum_{k=1}^n k^k\,$?

Defined $\,S_n=\sum_{k=1}^n k^k$, I am looking for a relation $\,S_{n+1}\sim f(n)\cdot S_n\,$, with $\,S_1=1$ and $f(n)\,$ a suitable function of $\,n$.

A trial and error procedure has led me to consider the following function: $$f(n)=\frac e 2\cdot\Big(2n+1-\log\Big(1+\frac{n-1}{(2n+1)^2}\Big)\Big)$$ Here are some experimental results:

$n\;\;\;\;\;\;\;\;\;\;err_\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n\;\;\;\;\;\;\;\;\;err_\%$

$1\;\;\;\;-18.45\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,20\;\;\;\;-1.33\%$

$2\;\;\;\;-14.09\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,30\;\;\;\;-0.85\%$

$3\;\;\;\;\;\,-9.70\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,40\;\;\;\;-0.61\%$

$4\;\;\;\;\;\,-7.17\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,50\;\;\;\;-0.47\%$

$5\;\;\;\;\;\,-5.68\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,60\;\;\;\;-0.37\%$

$6\;\;\;\;\;\,-4.70\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,70\;\;\;\;-0.30\%$

$7\;\;\;\;\;\,-4.00\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,80\;\;\;\;-0.25\%$

$8\;\;\;\;\;\,-3.49\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,90\;\;\;\;-0.21\%$

$9\;\;\;\;\;\,-3.08\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;100\;\;\;\;-0.18\%$

$10\;\;\,\;-2.76\%$

I shall appreciate every comment and suggestion. Many thanks.

Answer 1

$$S_n=\sum_{k=0}^{n}k^k$$ $$S_{n+1}^*=(2.71830821508223 n+1.3393100014856814) S_n^*;\;S_1=1$$ The general approximation is $$S_n^*=1.1286516133652116\cdot 0.367875870164984^{1- n} \;\Gamma (n+0.4926998322172109)$$

$$ \begin{array}{r|r|r|r} n & S_n & S_n^*& error ( \%)\\ \hline 1 & 1. & 1. & 0. \\ 2 & 5. & 4.05762 & 18.8476 \\ 3 & 32. & 27.4941 & 14.0809 \\ 4 & 288. & 261.036 & 9.36262 \\ 5 & 3413. & 3187.91 & 6.5951 \\ 6 & 50069. & 47598.2 & 4.9348 \\ 7 & 873612. & 840068. & 3.83968 \\ 8 & 1.76508\times 10^7 & 1.71101\times 10^7 & 3.06369 \\ 9 & 4.05071\times 10^8 & 3.94999\times 10^8 & 2.48655 \\ 10 & 1.04051\times 10^{10} & 1.01926\times 10^{10} & 2.04211 \\ 11 & 2.95717\times 10^{11} & 2.90717\times 10^{11} & 1.69072 \\ 12 & 9.21182\times 10^{12} & 9.0822\times 10^{12} & 1.40705 \\ 13 & 3.12087\times 10^{14} & 3.08423\times 10^{14} & 1.17414 \\ 14 & 1.14241\times 10^{16} & 1.13121\times 10^{16} & 0.980217 \\ 15 & 4.49318\times 10^{17} & 4.45648\times 10^{17} & 0.816844 \\ 16 & 1.88961\times 10^{19} & 1.8768\times 10^{19} & 0.677825 \\ 17 & 8.46136\times 10^{20} & 8.41411\times 10^{20} & 0.558506 \\ 18 & 4.01925\times 10^{22} & 4.00095\times 10^{22} & 0.455327 \\ 19 & 2.01861\times 10^{24} & 2.01123\times 10^{24} & 0.365523 \\ 20 & 1.06876\times 10^{26} & 1.0657\times 10^{26} & 0.286909 \\ \end{array} $$

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The formula is a simple linear regression on the ratios of $20$ sums $s_{k+1}/s_k$ They lie on a almost perfect straight line. I used $1000$ ratios to get the formulae above.

The picture below shows the first $20$

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