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A379152 The binary weights of the odd Catalan numbers.

%I A379152 #12 Dec 18 2024 03:15:47
%S A379152 1,1,2,6,16,25,60,127,244,494,1010,2015,4076,8086,16281,32818,65518,
%T A379152 131059,262348,524448,1047643,2097675,4194133,8386693,16776916,
%U A379152 33554390,67114125,134214652,268452748
%N A379152 The binary weights of the odd Catalan numbers.
%H A379152 Florian Luca and Paul Thomas Young, <a href="https://citeseerx.ist.psu.edu/document?repid=rep1&amp;type=pdf&amp;doi=44968aa10f5279654e257dbcb3a23c1b9c718053">On the binary expansion of the odd Catalan numbers</a>, Proceedings of the XIVth International Conference on Fibonacci Numbers, Morelia, Mexico, 2010, pp. 185-190; <a href="https://www.researchgate.net/publication/242547710_On_the_binary_expansion_of_the_odd_Catalan_numbers">ResearchGate link</a>.
%F A379152 a(n) = A000120(A038003(n)) = A000120(A000108(2^n-1)).
%F A379152 a(n) = A379151(2^n-1).
%t A379152 a[n_] := DigitCount[CatalanNumber[2^n-1], 2, 1]; Array[a, 23, 0]
%o A379152 (PARI) a(n) = my(m = -1 + 1 << n); hammingweight(binomial(2*m, m)/(m+1));
%o A379152 (Python)
%o A379152 from itertools import count, islice
%o A379152 def A379152_gen(): # generator of terms
%o A379152     yield from [1,1]
%o A379152     c, s = 1, 3
%o A379152     for n in count(2):
%o A379152         c = (c*((n<<2)-2))//(n+1)
%o A379152         if n == s:
%o A379152             yield c.bit_count()
%o A379152             s = (s<<1)|1
%o A379152 A379152_list = list(islice(A379152_gen(),10)) # _Chai Wah Wu_, Dec 17 2024
%Y A379152 Cf. A000108, A000120, A038003.
%Y A379152 Similar sequences: A011373, A079584, A082481, A379151, A379153.
%K A379152 nonn,base,more
%O A379152 0,3
%A A379152 _Amiram Eldar_, Dec 17 2024