A275012 Number of nonzero coefficients in the polynomial factor of the expression counting binomial coefficients with 2-adic valuation n.
1, 1, 4, 11, 29, 69, 174, 413, 995, 2364, 5581, 13082, 30600, 71111, 164660, 379682, 872749
Offset: 0
Examples
For n=2, the number of integers m such that binomial(k,m) is divisible by 2^n but not by 2^(n+1) is given by 2^X_1 (-1/8 X_10 + 1/8 X_10^2 + X_100 + 1/4 X_110), where X_w is the number of occurrences of the word w in the binary representation of k. The polynomial factor of this expression has a(2) = 4 nonzero terms. - _Eric Rowland_, Mar 05 2017
Links
- Eric Rowland, Binomial Coefficients, Valuations, and Words, In: Charlier É., Leroy J., Rigo M. (eds) Developments in Language Theory, DLT 2017, Lecture Notes in Computer Science, vol 10396.
- Lukas Spiegelhofer and Michael Wallner, An explicit generating function arising in counting binomial coefficients divisible by powers of primes, arXiv:1604.07089 [math.NT], 2016.
- Lukas Spiegelhofer and Michael Wallner, Divisibility of binomial coefficients by powers of two, arXiv:1710.10884 [math.NT], 2017.
Crossrefs
A001316, A163000, and A163577 count binomial coefficients with 2-adic valuation 0, 1, and 2. - Eric Rowland, Mar 15 2017
Extensions
a(12)-a(16) from Eric Rowland, Mar 20 2017