A321624 The Riordan square of the Lucas numbers, triangle read by rows, T(n, k) for 0 <= k <= n.

1, 1, 1, 3, 4, 1, 4, 10, 7, 1, 7, 24, 26, 10, 1, 11, 49, 77, 51, 13, 1, 18, 98, 200, 190, 85, 16, 1, 29, 187, 473, 595, 390, 128, 19, 1, 47, 350, 1056, 1658, 1450, 704, 180, 22, 1, 76, 642, 2253, 4255, 4688, 3062, 1159, 241, 25, 1

Offset: 0

Author

Peter Luschny, Nov 22 2018

Keywords

• nonn
• tabl

Comments

Compare A000032 (Lucas numbers with a(0) = 2), A000204 (Lucas numbers with a(0) undefined). Our variant has a(0) = 1.

Triangle, read by rows, given by [1, 2, -5/2, 1/2, 0, 0, 0, 0, 0, ...]DELTA[1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 06 2020

Examples

			[0] [  1]
[1] [  1,    1]
[2] [  3,    4,    1]
[3] [  4,   10,    7,    1]
[4] [  7,   24,   26,   10,    1]
[5] [ 11,   49,   77,   51,   13,    1]
[6] [ 18,   98,  200,  190,   85,   16,    1]
[7] [ 29,  187,  473,  595,  390,  128,   19,   1]
[8] [ 47,  350, 1056, 1658, 1450,  704,  180,  22,   1]
[9] [ 76,  642, 2253, 4255, 4688, 3062, 1159, 241,  25, 1]
		

Crossrefs

T(n, 0) = A000204, A000032 (Lucas), A321573 (row sums), A000007 (alternating row sums).

Cf. A321620.

Programs

Maple

# The function RiordanSquare is defined in A321620.
Lucas :=  1 + x*(1 + 2*x)/(1 - x - x^2); RiordanSquare(Lucas, 10);

Mathematica

(* The function RiordanSquare is defined in A321620. *)
Lucas = 1 + x*(1 + 2*x)/(1 - x - x^2);
RiordanSquare[Lucas, 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)

Sage

# uses[riordan_square from A321620]
riordan_square(1 + x*(1 + 2*x)/(1 - x - x^2), 10)

Formula

T(0,0) = 1, T(1,1) = 1, T(1,0) = 1, T(n,k) = 0 for k<0 and for k>n, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k) + 2*T(n-2,k-1), for n>1. - Philippe Deléham, Feb 06 2020