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

1,5

Comments

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

Companion to triangle A140071.

Links

Table of n, a(n) for n=1..66.

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)

Example

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;
  ...

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 *)

Crossrefs

Cf. A006012, A140071, A157751.

Sequence in context: A163366 A181145 A227203 * A158815 A101275 A262494

Adjacent sequences: A140067 A140068 A140069 * A140071 A140072 A140073

Keyword

nonn,tabl

Author

Gary W. Adamson, May 04 2008

Status

approved