A082481 - OEIS

%I #22 Dec 17 2024 08:45:47

%S 1,1,2,2,3,6,6,6,6,11,9,9,9,13,10,16,14,10,16,20,14,20,16,29,26,24,22,

%T 30,24,20,25,25,30,29,33,37,35,40,35,39,37,40,42,43,36,44,46,48,48,41,

%U 43,46,50,58,51,52,52,50,53,56,54,48,59,60,57,64,61,61,64,66,64,72,73

%N Number of 1's in binary representation of C(2n,n).

%H Robert Israel, <a href="/A082481/b082481.txt">Table of n, a(n) for n = 0..10000</a>

%H Amiram Eldar, <a href="/A082481/a082481.jpg">Plot of (1/n^2) * Sum_{k=1..n} a(k) for n = 2^(8..23)</a>.

%H Arnold Knopfmacher and Florian Luca, <a href="https://doi.org/10.1016/j.jnt.2011.07.004">Digit sums of binomial sums</a>, Journal of Number Theory, Vol. 132, No. 2 (2012), pp. 324-331.

%H Florian Luca and Igor E. Shparlinski, <a href="https://doi.org/10.1216/RMJ-2011-41-4-1291">On the g-ary expansions of middle binomial coefficients and Catalan numbers</a>, The Rocky Mountain Journal of Mathematics, Vol. 41, No. 4 (2011), pp. 1291-1301.

%F Should be asymptotic to n.

%F a(n) = A000120(A000984(n)). - _Michel Marcus_, Mar 27 2018

%F From _Amiram Eldar_, Dec 17 2024: (Start)

%F Two formulas from Luca and Shparlinski (2011):

%F a(n) >= 3 for all but finitely many positive integers n.

%F a(n) >> eps(n) * sqrt(log(n)), for all n <= X with at most o(X) exceptions as X -> oo, where eps(n) is a function tending to zero as n -> oo.

%F a(n) > c * log(n)/log(log(n)) holds on a set of n of asymptotic density 1, where c > 0 is a constant (Knopfmacher and Luca, 2012). (End)

%p seq(convert(convert(binomial(2*n,n),base,2),`+`),n=0..100); # _Robert Israel_, Mar 27 2018

%t Table[DigitCount[Binomial[2n,n],2,1],{n,0,90}] (* _Harvey P. Dale_, Jul 20 2023 *)

%o (PARI) a(n)=sum(k=1,length(binary(binomial(2*n,n))), component(binary(binomial(2*n,n)),k))

%o (PARI) a(n) = hammingweight(binomial(2*n, n)); \\ _Michel Marcus_, Mar 27 2018

%Y Cf. A000120, A000984.

%K nonn,base

%O 0,3

%A _Benoit Cloitre_, Apr 27 2003