And this works for any prime, not just 2. On Tue, Jan 26, 2021 at 5:33 PM Victor Miller <victorsmiller@gmail.com> wrote:
This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract.
On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://oeis.org/A001700>) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1.
Is this true for all n ≥ 0 ???
—Dan
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