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A321624
The Riordan square of the Lucas numbers, triangle read by rows, T(n, k) for 0 <= k <= n.
2
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
0,4
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
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
EXAMPLE
[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]
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 *)
PROG
(Sage) # uses[riordan_square from A321620]
riordan_square(1 + x*(1 + 2*x)/(1 - x - x^2), 10)
CROSSREFS
T(n, 0) = A000204, A000032 (Lucas), A321573 (row sums), A000007 (alternating row sums).
Cf. A321620.
Sequence in context: A247041 A299446 A300084 * A079529 A361508 A299022
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 22 2018
STATUS
approved