A321624 - OEIS

%I #19 Mar 27 2020 17:34:09

%S 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,

%T 16,1,29,187,473,595,390,128,19,1,47,350,1056,1658,1450,704,180,22,1,

%U 76,642,2253,4255,4688,3062,1159,241,25,1

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

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

%C 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

%F 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

%e [0] [  1]

%e [1] [  1,    1]

%e [2] [  3,    4,    1]

%e [3] [  4,   10,    7,    1]

%e [4] [  7,   24,   26,   10,    1]

%e [5] [ 11,   49,   77,   51,   13,    1]

%e [6] [ 18,   98,  200,  190,   85,   16,    1]

%e [7] [ 29,  187,  473,  595,  390,  128,   19,   1]

%e [8] [ 47,  350, 1056, 1658, 1450,  704,  180,  22,   1]

%e [9] [ 76,  642, 2253, 4255, 4688, 3062, 1159, 241,  25, 1]

%p # The function RiordanSquare is defined in A321620.

%p Lucas :=  1 + x*(1 + 2*x)/(1 - x - x^2); RiordanSquare(Lucas, 10);

%t (* The function RiordanSquare is defined in A321620. *)

%t Lucas = 1 + x*(1 + 2*x)/(1 - x - x^2);

%t RiordanSquare[Lucas, 10] (* _Jean-François Alcover_, Jun 15 2019, from Maple *)

%o (Sage) # uses[riordan_square from A321620]

%o riordan_square(1 + x*(1 + 2*x)/(1 - x - x^2), 10)

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

%Y Cf. A321620.

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Nov 22 2018