OFFSET
0,2
COMMENTS
Column k=0 yields A033543 (2nd binomial transform of the sequence A000957(n+1)). Row sums yield A133158. [Corrected by Philippe Deléham, Oct 24 2007, Dec 05 2009]
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) = 2*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 from 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 100 rows, flattened
FORMULA
T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k-1) for k >= 2.
Sum_{k=0..n} T(n,k)*(3*k+1) = 6^n. - Philippe Deléham, Mar 27 2007
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A033543(m+n). - Philippe Deléham, Nov 22 2009
EXAMPLE
Row 2 is (5,6,1) because M[3]= [2,1,0;1,4,1;0,1,4] and M[3]^2=[5,6,1;6,18,8;1,8,17].
Triangle starts:
1;
2, 1;
5, 6, 1;
16, 30, 10, 1;
62, 146, 71, 14, 1;
270, 717, 444, 128, 18, 1;
MAPLE
with(linalg): m:=proc(i, j) if i=1 and j=1 then 2 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; 2, 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} -> 2, 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