OFFSET
1,4
COMMENTS
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
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
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
FORMULA
Sum_{k=0..n} T(n,k)*(4*k+1) = 6^n. - Philippe Deléham, Mar 27 2007
EXAMPLE
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].
Triangle starts:
1;
1, 1;
2, 5, 1;
7, 23, 9, 1;
30, 108, 60, 13, 1;
138, 522, 361, 113, 17, 1;
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson & Roger L. Bagula, Nov 05 2006
EXTENSIONS
Edited by N. J. A. Sloane, Dec 04 2006
STATUS
approved