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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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%I #11 Nov 14 2023 04:44:39

%S 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,

%T 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,

%U 120,210,126,126,56,28,7,1

%N 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.

%C A140737 = triangle with reversed terms by rows. - _Gary W. Adamson_, May 25 2008

%C 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

%e First few rows of the triangle are:

%e 1;

%e 1, 1, 1;

%e 1, 1, 3, 2, 1;

%e 1, 1, 5, 4, 6, 3, 1;

%e 1, 1, 7, 6, 15, 10, 10, 4, 1;

%e 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1;

%e 1, 1, 11, 20, 45, 36, 84, 56, 70, 35, 21, 6, 1;

%e 1, 1, 13, 12, 66, 55, 165, 120, 210, 126, 126, 56, 28, 7, 1;

%e ...

%p A140736 := proc(n,k)

%p local X,r,c ;

%p X := Matrix(2*n+1,2*n+1) ;

%p for r from 1 to 2*n+1 do

%p for c from 1 to 2*n+1 do

%p if r = c then

%p if type(r,'odd') then

%p X[r,c] := 1 ;

%p else

%p X[r,c] := 0 ;

%p end if ;

%p elif r = c+1 or r=c+2 then

%p X[r,c] := 1 ;

%p end if;

%p end do:

%p end do:

%p LinearAlgebra[MatrixPower](X,n) ;

%p %[k,1] ;

%p end proc:

%p seq(seq( A140736(n,k),k=1..2*n+1),n=0..12) ; # _R. J. Mathar_, Nov 14 2023

%Y Cf. A001906 (row sums).

%Y Cf. A140737, A005408 (3rd column), A005843 (4th column), A000384 (5th column), A014105 (6th column), A000447 (7th column)

%K nonn,tabf,easy

%O 0,7

%A _Gary W. Adamson_ and _Roger L. Bagula_, May 25 2008