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Number of nonzero coefficients in the polynomial factor of the expression counting binomial coefficients with 2-adic valuation n.
3

%I #33 Jun 06 2021 09:05:57

%S 1,1,4,11,29,69,174,413,995,2364,5581,13082,30600,71111,164660,379682,

%T 872749

%N Number of nonzero coefficients in the polynomial factor of the expression counting binomial coefficients with 2-adic valuation n.

%H Eric Rowland, <a href="https://dx.doi.org/10.1007/978-3-319-62809-7_3">Binomial Coefficients, Valuations, and Words</a>, In: Charlier É., Leroy J., Rigo M. (eds) Developments in Language Theory, DLT 2017, Lecture Notes in Computer Science, vol 10396.

%H Lukas Spiegelhofer and Michael Wallner, <a href="https://arxiv.org/abs/1604.07089">An explicit generating function arising in counting binomial coefficients divisible by powers of primes</a>, arXiv:1604.07089 [math.NT], 2016.

%H Lukas Spiegelhofer and Michael Wallner, <a href="https://arxiv.org/abs/1710.10884">Divisibility of binomial coefficients by powers of two</a>, arXiv:1710.10884 [math.NT], 2017.

%e 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

%Y A001316, A163000, and A163577 count binomial coefficients with 2-adic valuation 0, 1, and 2. - _Eric Rowland_, Mar 15 2017

%K nonn,more

%O 0,3

%A _Michel Marcus_, Nov 12 2016

%E a(12)-a(16) from _Eric Rowland_, Mar 20 2017