The number of odd binomial coefficients in each row of Pascal's triangle is always a power of two although their sum rarely is. One of these rare occasions occurs for numbers of the form $\,$$n = 2^m -2$$\,$$\,$when the sum is exactly half of the total row sum. [One can see this e.g. by writing the binary expansion of$\,$ $n = 111...110$ $\,$and applying Lucas' classical result to conclude that, for numbers of this form, $\,$$ {n \choose k}$$\,$ is odd iff $\,$$k$$\,$ is even.]$\;$So this at least gives an infinite sequence of integers $\,$$2,6,14,30,...$$\;$which exhibit this odd behavior. Are there any other examples?

Question:$\;$If $\;$$\Sigma$${n\choose k}$ $=$$\,$$2^{n-1}$$\;$, where the sum extends over the odd binomial coefficients only, must $\,$$n$$\,$be of the form $\,$$2^m - 2$$\;$?


  • $\begingroup$ oeis.org/A088560 $\endgroup$ – mathlove Nov 28 '15 at 18:54
  • $\begingroup$ Thanks for the link. Unfortunately, the claims made there (although plausible) appear to be unsupported by any references or proofs. $\endgroup$ – user2052 Dec 8 '15 at 19:10

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