A335966 - OEIS

%I #24 Oct 08 2020 03:13:26

%S 1,2,2,2,2,6,2,2,2,4,4,4,4,14,2,2,2,4,4,4,4,12,4,4,4,8,8,8,8,30,2,2,2,

%T 4,4,4,4,12,4,4,4,8,8,8,8,28,4,4,4,8,8,8,8,24,8,8,8,16,16,16,16,62,2,

%U 2,2,4,4,4,4,12,4,4,4,8,8,8,8,28,4,4,4,8,8,8,8,24

%N a(n) is the number of odd terms in the n-th row of triangle A056939.

%C The entries of Baxter triangles are binomial(n+1, k-1)*binomial(n+1, k)*binomial(n+1, k+1)/(binomial(n+1, 1)*binomial(n+1, 2)).

%H Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, <a href="https://doi.org/10.1016/j.jcta.2010.03.017">Bijections for Baxter families and related objects</a>, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.

%F a(n) is even if n>=1.

%F a(n) = n iff n is of the form 2^k-2.

%F a(2^k-3) = 2^k-2.

%e a(4)=2 as there are two odd numbers among 1,10,10,1.

%t a[n_] := Count[Table[2 * Binomial[n, k] * Binomial[n + 1, k + 1] * Binomial[n + 2, k + 2]/((n - k + 1)^2 * (n - k + 2)), {k, 0, n}], _?OddQ]; Array[a, 100, 0] (* _Amiram Eldar_, Jul 02 2020 *)

%o (PARI) T(n,m) = 2*binomial(n, m)*binomial(n + 1, m + 1)*binomial(n + 2, m + 2)/(( n - m + 1)^2*(n - m + 2)); \\ A056939

%o a(n) = sum(m=0, n, T(n,m) % 2); \\ _Michel Marcus_, Jul 02 2020

%Y Cf. A056939.

%K nonn

%O 0,2

%A _Sen-Peng Eu_, Jul 01 2020