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A124575
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Triangle read by rows: row n is the first row of the matrix M[n]^(n-1), where M[n] is the n X n tridiagonal matrix with main diagonal (2,4,4,...) and super- and subdiagonals (1,1,1,...).
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29
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1, 2, 1, 5, 6, 1, 16, 30, 10, 1, 62, 146, 71, 14, 1, 270, 717, 444, 128, 18, 1, 1257, 3582, 2621, 974, 201, 22, 1, 6096, 18206, 15040, 6718, 1800, 290, 26, 1, 30398, 93960, 85084, 43712, 14208, 2986, 395, 30, 1, 154756, 491322, 478008, 274140, 103530
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OFFSET
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0,2
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COMMENTS
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Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 2*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) +y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
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LINKS
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FORMULA
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T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k-1) for k >= 2.
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EXAMPLE
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Row 2 is (5,6,1) because M[3]= [2,1,0;1,4,1;0,1,4] and M[3]^2=[5,6,1;6,18,8;1,8,17].
Triangle starts:
1;
2, 1;
5, 6, 1;
16, 30, 10, 1;
62, 146, 71, 14, 1;
270, 717, 444, 128, 18, 1;
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MAPLE
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with(linalg): m:=proc(i, j) if i=1 and j=1 then 2 elif i=j then 4 elif abs(i-j)=1 then 1 else 0 fi end: for n from 3 to 11 do A[n]:=matrix(n, n, m): B[n]:=multiply(seq(A[n], i=1..n-1)) od: 1; 2, 1; for n from 3 to 11 do seq(B[n][1, j], j=1..n) od; # yields sequence in triangular form
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MATHEMATICA
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M[n_] := SparseArray[{{1, 1} -> 2, Band[{2, 2}] -> 4, Band[{1, 2}] -> 1, Band[{2, 1}] -> 1}, {n, n}]; row[1] = {1}; row[n_] := MatrixPower[M[n], n-1] // First // Normal; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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