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%I #19 Apr 22 2017 04:40:03
%S 1,1,1,2,5,1,7,23,9,1,30,108,60,13,1,138,522,361,113,17,1,660,2587,
%T 2079,830,182,21,1,3247,13087,11733,5581,1579,267,25,1,16334,67328,
%U 65600,35636,12164,2672,368,29,1,83662,351246,365364,220308,86964,23220,4173
%N 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 (1,4,4,...) and super- and subdiagonals (1,1,1,...).
%C 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)=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
%C 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 by 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
%H G. C. Greubel, <a href="/A124576/b124576.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F Sum{k, 0<=k<=n} T(n,k)*(4*k+1) = 6^n. - _Philippe Deléham_, Mar 27 2007
%e Row 3 is (2,5,1) because M[3]=[1,1,0;1,4,1;0,1,4] and M[3]^2=[2,5,1;5,18,8;1,8,17].
%e Triangle starts:
%e 1;
%e 1, 1;
%e 2, 5, 1;
%e 7, 23, 9, 1;
%e 30, 108, 60, 13, 1;
%e 138, 522, 361, 113, 17, 1;
%p with(linalg): m:=proc(i,j) if i=1 and j=1 then 1 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; 1,1; for n from 3 to 11 do seq(B[n][1,j],j=1..n) od; # yields sequence in triangular form
%t M[n_] := SparseArray[{{1, 1} -> 1, 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 *)
%Y Cf. A124575, A124574, A052179.
%K nonn,tabl
%O 1,4
%A _Gary W. Adamson_ & _Roger L. Bagula_, Nov 05 2006
%E Edited by _N. J. A. Sloane_, Dec 04 2006
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