A140070 Triangle read by rows, iterates of matrix X * [1,0,0,0,...], where X = an infinite lower bidiagonal matrix with [1,3,1,3,1,3,...] in the main diagonal and [1,1,1,...] in the subdiagonal.

1, 1, 1, 1, 4, 1, 1, 13, 5, 1, 1, 40, 18, 8, 1, 1, 121, 58, 42, 9, 1, 1, 364, 179, 184, 51, 12, 1, 1, 1093, 543, 731, 235, 87, 13, 1, 1, 3280, 1636, 2736, 966, 496, 100, 16, 1, 1, 9841, 4916, 9844, 3702, 2454, 596, 148, 17, 1, 1, 29524, 14757, 34448, 13546, 11064, 3050, 1040, 165, 20, 1

Offset: 1

Author

Gary W. Adamson, May 04 2008

Keywords

• nonn
• tabl

Comments

Row sums = A006012: (1, 2, 6, 20, 68, 232, 792, 2704,...).

Companion to triangle A140071.

Examples

			First few rows of the triangle are:
  1;
  1,    1;
  1,    4,    1;
  1,   13,    5,    1;
  1,   40,   18,    8,   1;
  1,  121,   58,   42,   9,   1;
  1,  364,  179,  184,  51,  12,   1;
  1, 1093,  543,  731, 235,  87,  13,  1;
  1, 3280, 1636, 2736, 966, 496, 100, 16, 1;
  ...
		

Crossrefs

Cf. A006012, A140071, A157751.

Programs

Maple

T:= proc(n, k) option remember;
      `if`(k<0 or k>n, 0, `if`(k=0 or k=n, 1,
      4*T(n-1, k) - 3*T(n-2, k) + T(n-2, k-2)))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 18 2020

Mathematica

With[{m = 10}, CoefficientList[CoefficientList[Series[(1 + (y - 3)*x)/(1 - 4*x - (y^2 - 3)*x^2), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 18 2020 *)

Formula

Triangle read by rows, iterates of matrix X * [1,0,0,0,...], where X = an infinite lower bidiagonal matrix with [1,3,1,3,1,3,...] in the main diagonal and [1,1,1,...] in the subdiagonal; with the rest zeros.

From Peter Bala, Jan 17 2014: (Start)

O.g.f.: (1 + (x - 3)*z)/(1 - 4*z - (x^2 - 3)*z^2) = 1 + (x + 1)*z + (x^2 + 4*x + 1)*z^2 + ....

Recurrence equation: T(n,k) = 4*T(n-1,k) - 3*T(n-2,k) + T(n-2,k-2).

Recurrence equation for row polynomials: R(n,x) = 4*R(n-1,x) + (x^2 - 3)*R(n-2,x) with R(0,x) = 1 and R(1,x) = 1 + x.

Another recurrence equation: R(n,x) = (x + 2)*R(n-1,x) - R(n-1,-x) with R(0,x) = 1. Cf. A157751. (End)