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A140736 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.

%I A140736 #11 Nov 14 2023 04:44:39
%S A140736 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 A140736 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 A140736 120,210,126,126,56,28,7,1
%N A140736 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 A140736 A140737 = triangle with reversed terms by rows. - _Gary W. Adamson_, May 25 2008
%C A140736 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 A140736 First few rows of the triangle are:
%e A140736   1;
%e A140736   1, 1,  1;
%e A140736   1, 1,  3,  2,  1;
%e A140736   1, 1,  5,  4,  6,  3,   1;
%e A140736   1, 1,  7,  6, 15, 10,  10,   4,   1;
%e A140736   1, 1,  9,  8, 28, 21,  35,  20,  15,   5,   1;
%e A140736   1, 1, 11, 20, 45, 36,  84,  56,  70,  35,  21,  6,  1;
%e A140736   1, 1, 13, 12, 66, 55, 165, 120, 210, 126, 126, 56, 28, 7, 1;
%e A140736   ...
%p A140736 A140736 := proc(n,k)
%p A140736     local X,r,c ;
%p A140736     X := Matrix(2*n+1,2*n+1) ;
%p A140736     for r from 1 to 2*n+1 do
%p A140736     for c from 1 to 2*n+1 do
%p A140736         if r = c then
%p A140736             if type(r,'odd') then
%p A140736                 X[r,c] := 1 ;
%p A140736             else
%p A140736                 X[r,c] := 0 ;
%p A140736             end if ;
%p A140736         elif r = c+1 or r=c+2 then
%p A140736             X[r,c] := 1 ;
%p A140736         end if;
%p A140736     end do:
%p A140736     end do:
%p A140736     LinearAlgebra[MatrixPower](X,n) ;
%p A140736     %[k,1] ;
%p A140736 end proc:
%p A140736 seq(seq( A140736(n,k),k=1..2*n+1),n=0..12) ; # _R. J. Mathar_, Nov 14 2023
%Y A140736 Cf. A001906 (row sums).
%Y A140736 Cf. A140737, A005408 (3rd column), A005843 (4th column), A000384 (5th column), A014105 (6th column), A000447 (7th column)
%K A140736 nonn,tabf,easy
%O A140736 0,7
%A A140736 _Gary W. Adamson_ and _Roger L. Bagula_, May 25 2008