# Very odd binomial coefficients

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 $\,$$2^m - 2$$\;$?

Thanks

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