A228815 - OEIS

%I #24 Oct 12 2015 22:31:33

%S 0,1,1,1,2,1,2,5,5,2,3,10,14,10,3,5,20,36,36,20,5,8,38,83,106,83,38,8,

%T 13,71,182,281,281,182,71,13,21,130,382,690,834,690,382,130,21,34,235,

%U 778,1606,2268,2268,1606,778,235,34,55,420,1546,3586,5780,6750

%N Symmetric triangle, read by rows, related to Fibonacci numbers.

%C Triangles satisfying the same recurrence: A091533, A091562, A185081, A205575, A209137, A209138.

%F G.f.: x*(1+y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).

%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.

%F Sum_{k = 0..n} T(n,k)*x^k = A000045(n), 2*A015518(n), 3*A015524(n), 4*A200069(n) for x = 0, 1, 2, 3 respectively.

%F Sum_{k = 0..floor(n/2)} T(n-k,k) = A008998(n+1).

%e Triangle begins :

%e 0

%e 1, 1

%e 1, 2, 1

%e 2, 5, 5, 2

%e 3, 10, 14, 10, 3

%e 5, 20, 36, 36, 20, 5

%e 8, 38, 83, 106, 83, 38, 8

%e 13, 71, 182, 281, 281, 182, 71, 13

%e 21, 130, 382, 690, 834, 690, 382, 130, 21

%e 34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34

%e 55, 420, 1546, 3586, 5780, 6750, 5780, 3586, 1546, 420, 55

%Y Cf. A000045 (1st column), A001629 (2nd column), A008998, A152011, A261055 (3rd column).

%K nonn,easy,tabl

%O 0,5

%A _Philippe Deléham_, Oct 30 2013