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

0,3

Links

Table of n, a(n) for n=0..28.

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.

Formula

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

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000108, A000120, A038003.

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

Sequence in context: A071522 A192532 A395989 * A230853 A364666 A160620

Adjacent sequences: A379149 A379150 A379151 * A379153 A379154 A379155

Keyword

nonn,base,more

Author

Amiram Eldar, Dec 17 2024

Status

approved