A379152 The binary weights of the odd Catalan numbers.

1, 1, 2, 6, 16, 25, 60, 127, 244, 494, 1010, 2015, 4076, 8086, 16281, 32818, 65518, 131059, 262348, 524448, 1047643, 2097675, 4194133, 8386693, 16776916, 33554390, 67114125, 134214652, 268452748

Offset: 0

Author

Amiram Eldar, Dec 17 2024

Keywords

• nonn
• base
• more

Links

• Florian Luca and Paul Thomas Young, On the binary expansion of the odd Catalan numbers, Proceedings of the XIVth International Conference on Fibonacci Numbers, Morelia, Mexico, 2010, pp. 185-190; ResearchGate link.

Crossrefs

Cf. A000108, A000120, A038003.

Similar sequences: A011373, A079584, A082481, A379151, A379153.

Programs

Mathematica

a[n_] := DigitCount[CatalanNumber[2^n-1], 2, 1]; Array[a, 23, 0]

PARI

a(n) = my(m = -1 + 1 << n); hammingweight(binomial(2*m, m)/(m+1));

Python

from itertools import count, islice
def A379152_gen(): # generator of terms
    yield from [1,1]
    c, s = 1, 3
    for n in count(2):
        c = (c*((n<<2)-2))//(n+1)
        if n == s:
            yield c.bit_count()
            s = (s<<1)|1
A379152_list = list(islice(A379152_gen(),10)) # Chai Wah Wu, Dec 17 2024

Formula

a(n) = A000120(A038003(n)) = A000120(A000108(2^n-1)).

a(n) = A379151(2^n-1).