[seqfan] A surprising property of sums of binomial(n,i)

[Vladimir Shevelev]

Dear Seqfans,

Before submission sequences A258126, A258483, I naturally
believed that with the growth of n, the frequency of appearance
at least one prime of the form sum{0<=i<=k} biomial(n,i) , k=2,...,n-1,
increases.
However, studying the Peter's b-file for A258126 I noticed that
this frequency slowly decreases, and correspondingly the freaquency
of appearance of terms of A258483 slowly increases. I asked
Peter to write a table of change of the freaquency with step 100.
He get

The first few counts of terms of A258483 in steps of 100s

{22,56,89,131,170,209,253,292,337,373,420,469,511,566,597,642,687,734,783,823,860,918,963,1017,1065,1122,1172,1214,1266,1313,1365,1417,1461,1511,1562,1597,1649,1694,1737,1770,1797,1860,1918,1974,2025,2079,2133,2191,2255,2305}

Ratios:
0.22
0.28
0.296667
0.3275
0.34
0.348333
0.361429
0.365
0.374444
0.373
0.381818
0.390833
0.393077
0.404286
0.398
0.40125
0.404118
0.407778
0.412105
0.4115
0.409524
0.417273
0.418696
0.42375
0.426
0.431538
0.434074
0.433571
0.436552
0.437667
0.440323
0.442813
0.442727
0.444412
0.446286
0.443611
0.445676
0.445789
0.445385
0.4425
0.438293
0.442857
0.446047
0.448636
0.45
0.451957
0.45383
0.456458
0.460204
0.461

In whole the growth continues. Since the growth rate decreases, it indicates either to the limit or to the maximum (or small oscillations near a number).
In any case, it is a surprising property of rows of binomials which I still cannot explain.
If anyone can?

Best regards,
Vladimir