A124744 - OEIS

%I #14 Aug 27 2024 09:16:33

%S 1,0,1,0,0,1,0,0,-1,1,0,0,0,-1,1,0,0,0,0,-2,1,0,0,0,0,1,-2,1,0,0,0,0,

%T 0,1,-3,1,0,0,0,0,0,0,3,-3,1,0,0,0,0,0,0,-1,3,-4,1,0,0,0,0,0,0,0,-1,6,

%U -4,1,0,0,0,0,0,0,0,0,-4,6,-5,1,0,0,0,0,0,0,0,0,1,-4,10,-5,1

%N Expansion of (1+x*y)/(1-x^2*y^2+x^3*y^2).

%H Paolo Xausa, <a href="/A124744/b124744.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).

%F T(n,k) = binomial(floor(k/2),n-k)*(-1)^(n-k)

%F Column k has g.f. x^k*(1-x)^floor(k/2). - _Paul Barry_, Feb 01 2007

%e Triangle begins

%e   1,

%e   0, 1,

%e   0, 0, 1,

%e   0, 0, -1, 1,

%e   0, 0, 0, -1, 1,

%e   0, 0, 0, 0, -2, 1,

%e   0, 0, 0, 0, 1, -2, 1,

%e   0, 0, 0, 0, 0, 1, -3, 1,

%e   0, 0, 0, 0, 0, 0, 3, -3, 1,

%e   0, 0, 0, 0, 0, 0, -1, 3, -4, 1,

%e   0, 0, 0, 0, 0, 0, 0, -1, 6, -4, 1

%t Table[(-1)^(n-k)*Binomial[Floor[k/2], n-k], {n, 0, 15}, {k, 0, n}] (* _Paolo Xausa_, Aug 27 2024 *)

%Y Cf. A124745 (row sums), A124746 (diagonal sums), A124747 (inverse).

%K easy,sign,tabl

%O 0,20

%A _Paul Barry_, Nov 06 2006