OFFSET
0,2
COMMENTS
Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 4*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k >= 1.
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
From R. J. Mathar, Mar 12 2013: (Start)
The matrix inverse starts
1;
-4, 1;
11, -7, 1;
-29, 31, -10, 1;
76, -115, 60, -13, 1;
-199, 390, -285, 98, -16, 1;
521, -1254, 1185, -566, 145, -19, 1;
-1364, 3893, -4524, 2785, -985, 201, -22, 1; ... (End)
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
FORMULA
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A026378(m+n+1).
Sum_{k=0..n} T(n,k) = 5^n = A000351(n).
T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,3/2) - GegenbauerC(n-k-1,-n+1,3/2)). - Peter Luschny, May 13 2016
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 + x )*(1 + 3*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022
EXAMPLE
Triangle begins:
1;
4, 1;
17, 7, 1;
75, 39, 10, 1;
339, 202, 70, 13, 1;
1558, 1015, 425, 110, 16, 1;
7247, 5028, 2400, 771, 159, 19, 1;
34016, 24731, 12999, 4872, 1267, 217, 22, 1; ...
From Philippe Deléham, Nov 07 2011: (Start)
Production matrix begins:
4, 1
1, 3, 1
0, 1, 3, 1
0, 0, 1, 3, 1
0, 0, 0, 1, 3, 1
0, 0, 0, 0, 1, 3, 1
0, 0, 0, 0, 0, 1, 3, 1
0, 0, 0, 0, 0, 0, 1, 3, 1
0, 0, 0, 0, 0, 0, 0, 1, 3, 1 (End)
MAPLE
A126791 := proc(n, k)
if n=0 and k = 0 then
1 ;
elif k <0 or k>n then
0;
elif k= 0 then
4*procname(n-1, 0)+procname(n-1, 1) ;
else
procname(n-1, k-1)+3*procname(n-1, k)+procname(n-1, k+1) ;
end if;
end proc: # R. J. Mathar, Mar 12 2013
T := (n, k) -> (-1)^(n-k)*simplify(GegenbauerC(n-k, -n+1, 3/2) - GegenbauerC(n-k-1, -n+1, 3/2)): seq(seq(T(n, k), k=1..n), n=1..10); # Peter Luschny, May 13 2016
MATHEMATICA
T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,
T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 4, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Mar 14 2007
STATUS
approved