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Table T(n,k) = -2*T(n-1,k)+T(n-1,k+1) = T(n,k-4), 0<=n.
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%I #10 Sep 29 2024 16:04:04

%S 0,0,1,-1,0,1,-3,2,1,-5,8,-4,-7,18,-20,9,32,-56,49,-25,-120,161,-123,

%T 82,401,-445,328,-284,-1247,1218,-940,969,3712,-3376,2849,-3185,

%U -10800,9601,-8883,10082,31201,-28085,27848,-30964,-90487,84018,-86660,93129,264992,-254696

%N Table T(n,k) = -2*T(n-1,k)+T(n-1,k+1) = T(n,k-4), 0<=n.

%C The table is created from a nucleus (0,0,1,-1) in the upper row, periodic in each row with length 4 and extended downwards to further rows with the Pascal/Galton mixing coefficients (0,-2,1).

%C Each row may be regarded as coefficients in recurrences of a group of sequences, b_i(n) = Sum_{k=1..4} T(i,k)*b_i(n-k).

%F T(n,k) = T(n,k-1)+T(n,k-2)+T(n,k-3)+2*T(n,k-4).

%F Row sums are Sum_{k=1..4} T(n,k) = 0.

%F Row sums of absolute values: Sum_{k=1..4} |T(n,k)| = A008776(n).

%F A140289(n)=T(n,1). A140290(n)=T(n,4).

%e The table starts:

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

%e 0, 1, -3, 2, 0, 1, -3, 2,...

%e 1, -5, 8, -4, 1, -5, 8, -4,...

%e -7, 18, -20, 9, -7, 18, -20, 9,...

%e and only the first 4 columns (the non-redundant information) build the sequence.

%K sign,tabf,less

%O 0,7

%A _Paul Curtz_, May 24 2008

%E Edited and extended by _R. J. Mathar_, Jul 22 2008