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A379151 The binary weights of the Catalan numbers (A000108).

%I A379151 #16 Feb 05 2025 14:18:58
%S A379151 1,1,1,2,3,3,2,6,6,9,6,8,8,12,9,16,13,17,12,17,13,18,15,25,20,17,20,
%T A379151 24,28,25,26,25,25,32,27,34,29,32,33,29,35,29,31,36,35,44,44,49,40,46,
%U A379151 48,44,38,50,43,44,46,47,55,50,52,58,59,60,65,68,56,62,68
%N A379151 The binary weights of the Catalan numbers (A000108).
%H A379151 Amiram Eldar, <a href="/A379151/b379151.txt">Table of n, a(n) for n = 0..10000</a>
%H A379151 Amiram Eldar, <a href="/A379151/a379151.jpg">Plot of (1/n^2) * Sum_{k=1..n} a(k) for n = 2^(8..23)</a>.
%H A379151 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 A379151 a(n) = A000120(A000108(n)).
%F A379151 Two formulas from Luca and Shparlinski (2011):
%F A379151 a(n) >= 3 for all but finitely many positive integers n.
%F A379151 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 A379151 Conjecture: Sum_{k=1..n} a(k) ~ n^2 / 2 (see the plot in the Links section).
%e A379151 a(10) = 6 because Catalan(10) = 16796 = 100000110011100_2, which has 6 one bits. - _Vincenzo Librandi_, Feb 05 2025
%t A379151 a[n_] := DigitCount[CatalanNumber[n], 2, 1]; Array[a, 100, 0]
%o A379151 (PARI) a(n) = hammingweight(binomial(2*n, n)/(n+1));
%o A379151 (Magma) [&+Intseq(Catalan(n), 2): n in [0..100]]; // _Vincenzo Librandi_, Feb 05 2025
%Y A379151 Cf. A000102, A000108.
%Y A379151 Similar sequences: A011373, A079584, A082481, A379152, A379153.
%K A379151 nonn,easy,base
%O A379151 0,4
%A A379151 _Amiram Eldar_, Dec 16 2024