A191582 - OEIS

%I #11 Sep 04 2022 10:47:51

%S 1,0,1,3,1,1,0,4,2,1,9,4,6,3,1,0,13,10,9,4,1,27,13,23,19,13,5,1,0,40,

%T 36,42,32,18,6,1,81,40,76,78,74,50,24,7,1,0,121,116,154,152,124,74,31,

%U 8,1,243,121,237,270,306,276,198,105,39,9,1,0,364,358,507,576,582,474,303,144,48,10,1,729,364,722,865,1083,1158,1056,777,447,192,58,11,1

%N Riordan matrix (1/(1-3*x^2),x/(1-x)).

%C Row sums = A167936(n+1).

%C Diagonal sums = A191584.

%C Central coefficients = A191585.

%C Alternated row sums: Sum_{k=0..n} (-1)^(n-k)*T(n,k) = 3^floor(n/2) (A167936).

%C Binomial row sums: Sum_{k=0..n} binomial(n,k)*T(n,k) = central coefficients.

%F T(n,k) = Sum_{i=0..(n-k)/2} binomial(n-2*i-1,n-k-2*i)*3^i.

%F Recurrence: T(n+1,k+1) = T(n,k) + T(n,k+1).

%e Triangle begins:

%e 1

%e 0, 1

%e 3, 1, 1

%e 0, 4, 2, 1

%e 9, 4, 6, 3, 1

%e 0, 13, 10, 9, 4, 1

%e 27, 13, 23, 19, 13, 5, 1

%e 0, 40, 36, 42, 32, 18, 6, 1

%e 81, 40, 76, 78, 74, 50, 24, 7, 1

%t Flatten[Table[Sum[Binomial[n-2i-1,n-k-2i]3^i,{i,0,((n-k))/2}],{n,0,20},{k,0,n}]]

%o (Maxima) create_list(sum(binomial(n-2*i-1,n-k-2*i)*3^i,i,0,(n-k)/2),n,0,20,k,0,n);

%Y Cf. A167936, A191584, A191585.

%K nonn,easy,tabl

%O 0,4

%A _Emanuele Munarini_, Jun 07 2011