A180165 Triangle read by rows, derived from an array of sequences generated from (1 + x)/ (1 - r*x - r*x^2).

1, 1, 2, 1, 3, 3, 1, 4, 8, 5, 1, 5, 15, 22, 8, 1, 6, 24, 57, 60, 13, 1, 7, 35, 116, 216, 164, 21, 1, 8, 48, 205, 560, 819, 448, 34, 1, 9, 63, 330, 1200, 2704, 3105, 1224, 55, 1, 10, 80, 497, 2268, 7025, 13056, 11772, 3344, 89, 1, 11, 99, 712, 3920, 15588, 41125, 63040, 44631, 9136, 144

Offset: 1

Author

Gary W. Adamson, Aug 14 2010

Keywords

• nonn
• tabl

Comments

Row sums = A180166: (1, 3, 7, 18, 51, 161, 560, 2163, ...).

Rows of the array, with other offsets: (row 1 = A000045 starting with offset 2: (1, 2, 3, 5, 8, 13, ...); and for rows > 1, the entries: A028859, A125145, A086347, and A180033 start with offset 0; with the offset in the present array = 1.

Examples

			First few rows of the triangle:
  1;
  1, 2;
  1, 3, 3;
  1, 4, 8, 5;
  1, 5, 15, 22, 8;
  1, 6, 24, 57, 60, 13;
  1, 7, 35, 116, 216, 164, 21;
  1, 8, 48, 205, 560, 819, 448, 34;
  1, 9, 63, 330, 1200, 2704, 3105, 1224, 55;
  1, 10, 80, 497, 2268, 7025, 13056, 11772, 3344, 89;
  1, 11, 99, 712, 3920, 15588, 41125, 63040, 44631, 9136, 144;
  1, 12, 120, 981, 6336, 30919, 107136, 240750, 304384, 169209, 24960, 233;
  ...
As an array A(r,k) by upwards antidiagonals:
        k=1  k=2  k=3   k=4    k=5
  r=1:   1,   2,    3,    5,     8, ...
  r=2:   1,   3,    8,   22,    60, ...
  r=3:   1,   4,   15,   57,   216, ...
  r=4:   1,   5,   24,  116,   560, ...
  r=5:   1,   6,   35,  205,  1200, ...
Row r=5 = A180033 = (1, 6, 35, 205,...) and is generated from (1+x)/(1-5*x-5*x^2); is the INVERT transform of row r=4; and the array term A(5,4) = 205 = 5*35 + 5*6.
Terms A(2,4) and A(2,5) = [22,60] = [0,1; 2,2]^3 * [1,3].
		

Links

• Robert P. P. McKone, Antidiagonals n = 0..199, flattened

Crossrefs

Cf. A180166, A000045, A028859, A125145, A086347, A180033.

Cf. A340156.

Programs

Mathematica

A180165[a_] := Reverse[Table[Table[CoefficientList[Series[(1 + x)/(1 - r*x - r*x^2), {x, 0, a - 2}], x], {r, 1, a + 1}][[k, n - k]], {n, 1, a}, {k, 1, n - 1}], 2] // Flatten;
A180165[12] (* Robert P. P. McKone, Jan 19 2021 *)

Formula

Triangle read by rows, generated from an array of sequences generated from (1 + x)/(1 - r*x - r*x^2); r > 0.

Alternatively, given the array with offset 1, the sequence r-th sequence is generated from a(k) = r*a(k-1) + r*(k-2); a(1) = 1, a(2) = r+1.

With a matrix method, the array can be generated from a 2 X 2 matrix of the form [0,1; r,r] = M, such that M^n * [1,r+1] = [r,n+1; r,n+2].

Also, for r > 1, the (r+1)-th row of the array is the INVERT transform of the r-th row.

Extensions

a(35) corrected by Robert P. P. McKone, Dec 31 2020