A329723 -id:A329723

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A329722

a(n) = Sum_{k=0..n} ((binomial(n+2k,2n-k)*binomial(n,k)) mod 2).

+10
2

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 4, 1, 1, 3, 4, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 2, 4, 2, 2, 4, 2, 1, 1, 1, 2, 3, 3, 4, 7, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 4, 1, 1

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OFFSET

0,4

COMMENTS

Run length transform of the coefficients of (1-2x^3)/(1-x-x^2), i.e., 1, 1, 2, 1, 3, 4, 7, 11, ... (1, 1 followed by the Lucas sequence A000032).

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

PROG

(PARI) a(n) = sum(k=0, n, lift(Mod((binomial(n+2*k, 2*n-k)*binomial(n, k)), 2))) \\ Felix Fröhlich, Nov 25 2019

(Python)

def A329722(n): return sum(int(not (~(n+2*k) & 2*n-k) | (~n & k)) for k in range(n+1)) # Chai Wah Wu, Sep 28 2021

CROSSREFS

Cf. A000032, A329723.

KEYWORD

nonn

AUTHOR

Chai Wah Wu, Nov 19 2019

STATUS

approved

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