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

0,3

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.

Formula

Should be asymptotic to n.

a(n) = A000120(A000984(n)). - Michel Marcus, Mar 27 2018

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)

Maple

seq(convert(convert(binomial(2*n, n), base, 2), `+`), n=0..100); # Robert Israel, Mar 27 2018

Mathematica

Table[DigitCount[Binomial[2n, n], 2, 1], {n, 0, 90}] (* Harvey P. Dale, Jul 20 2023 *)

Prog

(PARI) a(n)=sum(k=1, length(binary(binomial(2*n, n))), component(binary(binomial(2*n, n)), k))
(PARI) a(n) = hammingweight(binomial(2*n, n)); \\ Michel Marcus, Mar 27 2018

Crossrefs

Cf. A000120, A000984.

Sequence in context: A068424 A298484 A139359 * A136573 A121457 A093784

Adjacent sequences: A082478 A082479 A082480 * A082482 A082483 A082484

Keyword

nonn,base

Author

Benoit Cloitre, Apr 27 2003

Status

approved