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A088961 Zigzag matrices listed entry by entry.

%I A088961 #15 Aug 19 2015 11:59:04
%S A088961 3,5,5,5,10,14,14,7,14,21,21,7,21,35,42,48,27,9,48,69,57,36,27,57,78,
%T A088961 84,9,36,84,126,132,165,110,44,11,165,242,209,121,55,110,209,253,220,
%U A088961 165,44,121,220,297,330,11,55,165,330,462
%N A088961 Zigzag matrices listed entry by entry.
%C A088961 For each n >= 1 the n X n matrix Z(n) is constructed as follows. The i-th row of Z(n) is obtained by generating a hexagonal array of numbers with 2*n+1 rows, 2*n numbers in the odd numbered rows and 2*n+1 numbers in the even numbered rows. The first row is all 0's except for two 1's in the i-th and the (2*n+1-i)th positions. The remaining rows are generated using the same rule for generating Pascal's triangle. The i-th row of Z(n) then consists of the first n numbers in the bottom row of our array.
%C A088961 For example the top row of Z(2) is [5,5], found from the array:
%C A088961 . 1 0 0 1
%C A088961 1 1 0 1 1
%C A088961 . 2 1 1 2
%C A088961 2 3 2 3 2
%C A088961 . 5 5 5 5
%C A088961 Zigzag matrices have remarkable properties. Here is a selection:
%C A088961 1) Z(n) is symmetric.
%C A088961 2) det(Z(n)) = A085527(n).
%C A088961 3) tr(Z(n)) = A033876(n-1).
%C A088961 4) If 2*n+1 is a power of a prime p then all entries of Z(n) are multiples of p.
%C A088961 5) If 4*n+1 is a power of a prime p then the dot product of any two distinct rows of Z(n) is a multiple of p.
%C A088961 6) It is always possible to move from the bottom left entry of Z(n) to the top right entry using only rightward and upward moves and visiting only odd numbers.
%C A088961 A001700(n) = last term of last row of Z(n): a(A000330(n-1)) = A001700(n); A230585(n) = first term of first row of Z(n): a(A056520(n-1)) = A230585(n); A051417(n) = greatest common divisor of entries of Z(n). - _Reinhard Zumkeller_, Oct 25 2013
%H A088961 Reinhard Zumkeller, <a href="/A088961/b088961.txt">Matrices Z(n): n = 1..30, flattened</a>
%F A088961 The ij entry of Z(n) is binomial(2*n, n+j-i) - binomial(2*n, n+i+j) + binomial(2*n, 3*n+1-i-j).
%e A088961 The first five values are 3, 5, 5, 5, 10 because the first two zigzag matrices are [[3]] and [[5,5],[5,10]].
%t A088961 Flatten[Table[Binomial[2n,n+j-i]-Binomial[2n,n+i+j]+ Binomial[2n, 3n+1-i-j], {n,5},{i,n},{j,n}]] (* _Harvey P. Dale_, Dec 15 2011 *)
%o A088961 (Haskell)
%o A088961 a088961 n = a088961_list !! (n-1)
%o A088961 a088961_list = concat $ concat $ map f [1..] where
%o A088961    f x = take x $ g (take x (1 : [0,0..])) where
%o A088961      g us = (take x $ g' us) : g (0 : init us)
%o A088961      g' vs = last $ take (2 * x + 1) $
%o A088961                     map snd $ iterate h (0, vs ++ reverse vs)
%o A088961    h (p,ws) = (1 - p, drop p $ zipWith (+) ([0] ++ ws) (ws ++ [0]))
%o A088961 -- _Reinhard Zumkeller_, Oct 25 2013
%Y A088961 Cf. A085527, A003876.
%K A088961 easy,nice,nonn,look
%O A088961 1,1
%A A088961 _Paul Boddington_, Oct 28 2003