A126791 Binomial matrix applied to A111418.

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, 160795, 121208, 68600, 28882, 8890, 1940, 284, 25, 1, 764388, 593019, 355890, 164136

Offset: 0

Author

Philippe Deléham, Mar 14 2007

Keywords

• nonn
• tabl

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)

Examples

			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)
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Programs

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 *)

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