A277561 a(n) = Sum_{k=0..n} ({binomial(n+2k,2k)*binomial(n,k)} mod 2).

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 4, 4, 2, 4, 2, 2, 2, 4, 4, 4, 4, 8, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 2, 4, 4, 4, 4, 8, 4, 4, 4, 8, 8, 8, 4, 8, 4, 4, 2, 4, 4, 4, 4, 8, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 2, 4, 4, 4, 4, 8, 4, 4, 4, 8, 8, 8, 4, 8, 4, 4, 4, 8, 8, 8, 8, 16, 8

Offset: 0

Author

Chai Wah Wu, Oct 19 2016

Keywords

• nonn

Comments

Equals the run length transform of A040000: 1,2,2,2,2,2,...

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.
• Index entries for sequences computed with run length transform

Crossrefs

Cf. A005940, A034444, A040000, A069010, A106737.

Programs

Mathematica

Table[Sum[Mod[Binomial[n + 2 k, 2 k] Binomial[n, k], 2], {k, 0, n}], {n, 0, 86}] (* Michael De Vlieger, Oct 21 2016 *)

PARI

a(n) = sum(k=0, n, binomial(n+2*k, 2*k)*binomial(n,k) % 2); \\ Michel Marcus, Oct 21 2016

Python

def A277561(n):
    return sum(int(not (~(n+2*k) & 2*k) | (~n & k)) for k in range(n+1))

Formula

a(n) = 2^A069010(n). a(2n) = a(n), a(4n+1) = 2a(n), a(4n+3) = a(2n+1). - Chai Wah Wu, Nov 04 2016

a(n) = A034444(A005940(1+n)). - Antti Karttunen, May 29 2017