A082481 -id:A082481 - cp's OEIS frontend

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

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, 24, 28, 25, 26, 25, 25, 32, 27, 34, 29, 32, 33, 29, 35, 29, 31, 36, 35, 44, 44, 49, 40, 46, 48, 44, 38, 50, 43, 44, 46, 47, 55, 50, 52, 58, 59, 60, 65, 68, 56, 62, 68

Offset: 0

Amiram Eldar, Dec 16 2024

			a(10) = 6 because Catalan(10) = 16796 = 100000110011100_2, which has 6 one bits. - _Vincenzo Librandi_, Feb 05 2025

Amiram Eldar, 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).
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.

Cf. A000102, A000108.

Similar sequences: A011373, A079584, A082481, A379152, A379153.

[&+Intseq(Catalan(n), 2): n in [0..100]]; // Vincenzo Librandi, Feb 05 2025

a[n_] := DigitCount[CatalanNumber[n], 2, 1]; Array[a, 100, 0]

a(n) = hammingweight(binomial(2*n, n)/(n+1));

a(n) = A000120(A000108(n)).
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.
Conjecture: Sum_{k=1..n} a(k) ~ n^2 / 2 (see the plot in the Links section).

A379153 The binary weights of the Apéry numbers (A005259).

Original entry on oeis.org

1, 2, 3, 6, 6, 14, 15, 15, 20, 19, 23, 23, 27, 34, 35, 44, 40, 36, 40, 44, 41, 48, 52, 62, 64, 66, 57, 66, 72, 79, 71, 75, 77, 78, 79, 78, 88, 86, 92, 100, 103, 103, 92, 116, 96, 116, 117, 113, 129, 117, 123, 128, 123, 126, 130, 133, 129, 142, 147, 134, 135, 148

Offset: 0

Amiram Eldar, Dec 17 2024

Amiram Eldar, Table of n, a(n) for n = 0..10000
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 Apéry, Motzkin, Schröder and other combinatorial numbers, Annals of Combinatorics, Vol. 14 (2010), pp. 507-524.

Cf. A000120, A005259.

Similar sequences: A011373, A079584, A082481, A379151, A379152.

a[n_] := DigitCount[Sum[(Binomial[n, k] * Binomial[n+k, k])^2, {k, 0, n}], 2, 1]; Array[a, 100, 0]

a(n) = hammingweight(sum(k=0, n, (binomial(n, k)*binomial(n+k, k))^2));

a(n) = A000120(A005259(n)).
a(n) > c * (log(n)/log(log(n)))^(1/4) holds on a set of n of asymptotic density 1, where c > 0 is a constant (Luca and Shparlinski, 2010).
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).
Conjecture: Limit_{m->oo} (1/m^2) * Sum_{k=1..m} a(k) = log(sqrt(2) + 1)/log(2) = 1.2715533... (Knopfmacher and Luca, 2012).

A379152 The binary weights of the odd Catalan numbers.

Original entry on oeis.org

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

Amiram Eldar, Dec 17 2024

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.

Cf. A000108, A000120, A038003.

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

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

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

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

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

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

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

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A379153 The binary weights of the Apéry numbers (A005259).

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A379152 The binary weights of the odd Catalan numbers.

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