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A140736
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Triangle read by rows, X^n * [1,0,0,0,...]; where X = a tridiagonal matrix with (1,0,1,0,1,...) in the main diagonal and (1,1,1,...) in the sub- and subsubdiagonals.
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3
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1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 5, 4, 6, 3, 1, 1, 1, 7, 6, 15, 10, 10, 4, 1, 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1, 1, 1, 11, 10, 45, 36, 84, 56, 70, 35, 21, 6, 1, 1, 1, 13, 12, 66, 55, 165, 120, 210, 126, 126, 56, 28, 7, 1
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OFFSET
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0,7
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COMMENTS
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T(n,k) is the element in column 1 of row k of the n-th power of the (2n+1)X(2n+1) tridiagonal matrix X with X(r,c) = 1 if (r=c and r odd) or r=c+1 or r=c+2. - R. J. Mathar, Nov 14 2023
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LINKS
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EXAMPLE
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First few rows of the triangle are:
1;
1, 1, 1;
1, 1, 3, 2, 1;
1, 1, 5, 4, 6, 3, 1;
1, 1, 7, 6, 15, 10, 10, 4, 1;
1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1;
1, 1, 11, 20, 45, 36, 84, 56, 70, 35, 21, 6, 1;
1, 1, 13, 12, 66, 55, 165, 120, 210, 126, 126, 56, 28, 7, 1;
...
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MAPLE
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local X, r, c ;
X := Matrix(2*n+1, 2*n+1) ;
for r from 1 to 2*n+1 do
for c from 1 to 2*n+1 do
if r = c then
if type(r, 'odd') then
X[r, c] := 1 ;
else
X[r, c] := 0 ;
end if ;
elif r = c+1 or r=c+2 then
X[r, c] := 1 ;
end if;
end do:
end do:
LinearAlgebra[MatrixPower](X, n) ;
%[k, 1] ;
end proc:
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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