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A scaled Legendre triangle.
4

%I #6 Mar 18 2017 23:29:28

%S 1,0,1,-2,2,1,0,4,4,1,6,8,22,6,1,0,16,136,52,8,1,-20,32,886,504,94,10,

%T 1,0,64,5944,5136,1232,148,12,1,70,128,40636,53856,16966,2440,214,14,

%U 1,0,256,281488,575296,240368,42256,4248,292,16,1,-252,512,1968934,6225792,3468844,752800,88566,6776,382,18,1

%N A scaled Legendre triangle.

%C Row sums are A110125. Diagonal sums are A110126. Columns include A000079, A069835, A084773, and A098269.

%F Number triangle T(n, k)=2^(n-k)*LegendreP(n-k, k); T(n, k)=sum{j=0..floor((n-k)/2), (-1)^j*C(n-k, j)C(2n-2k-2j, n-k)k^(n-k-2j)}.

%e Rows begin

%e 1;

%e 0,1;

%e -2,2,1;

%e 0,4,4,1;

%e 6,8,22,6,1;

%e 0,16,136,62,8,1;

%e -20,32,886,504,94,10,1;

%o (PARI) T(n, k)=pollegendre(n-k, k)<<(n-k) \\ _Charles R Greathouse IV_, Mar 18 2017

%K easy,sign,tabl

%O 0,4

%A _Paul Barry_, Jul 13 2005