A082481 Number of 1's in binary representation of C(2n,n).
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, 30, 24, 20, 25, 25, 30, 29, 33, 37, 35, 40, 35, 39, 37, 40, 42, 43, 36, 44, 46, 48, 48, 41, 43, 46, 50, 58, 51, 52, 52, 50, 53, 56, 54, 48, 59, 60, 57, 64, 61, 61, 64, 66, 64, 72, 73
Offset: 0
Links
- Robert Israel, Table of n, a(n) for n = 0..10000
- Amiram Eldar, Plot of (1/n^2) * Sum_{k=1..n} a(k) for n = 2^(8..23).
- Arnold Knopfmacher and Florian Luca, Digit sums of binomial sums, Journal of Number Theory, Vol. 132, No. 2 (2012), pp. 324-331.
- Florian Luca and Igor E. Shparlinski, On the g-ary expansions of middle binomial coefficients and Catalan numbers, The Rocky Mountain Journal of Mathematics, Vol. 41, No. 4 (2011), pp. 1291-1301.
Programs
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Maple
seq(convert(convert(binomial(2*n,n),base,2),`+`),n=0..100); # Robert Israel, Mar 27 2018
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Mathematica
Table[DigitCount[Binomial[2n,n],2,1],{n,0,90}] (* Harvey P. Dale, Jul 20 2023 *) -
PARI
a(n)=sum(k=1,length(binary(binomial(2*n,n))), component(binary(binomial(2*n,n)),k))
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PARI
a(n) = hammingweight(binomial(2*n, n)); \\ Michel Marcus, Mar 27 2018
Formula
Should be asymptotic to n.
From Amiram Eldar, Dec 17 2024: (Start)
Two formulas from Luca and Shparlinski (2011):
a(n) >= 3 for all but finitely many positive integers n.
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.
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)