A321622 The Riordan square of the Fine numbers, triangle read by rows, T(n, k) for 0 <= k<= n.

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 6, 10, 7, 3, 1, 1, 18, 31, 19, 10, 4, 1, 1, 57, 97, 61, 29, 13, 5, 1, 1, 186, 316, 196, 96, 40, 16, 6, 1, 1, 622, 1054, 652, 316, 136, 52, 19, 7, 1, 1, 2120, 3586, 2210, 1072, 458, 181, 65, 22, 8, 1, 1

Offset: 0

Author

Peter Luschny, Nov 22 2018

Keywords

• nonn
• tabl

Comments

Fine numbers as defined in A000957 have a(0) = 0 whereas our variant has a(0) = 1. The rows sums of the triangle are |A002420|.

Examples

			[0] [    1]
[1] [    1,     1]
[2] [    0,     1,     1]
[3] [    1,     1,     1,     1]
[4] [    2,     4,     2,     1,     1]
[5] [    6,    10,     7,     3,     1,     1]
[6] [   18,    31,    19,    10,     4,     1,   1]
[7] [   57,    97,    61,    29,    13,     5,   1,  1]
[8] [  186,   316,   196,    96,    40,    16,   6,  1,  1]
[9] [  622,  1054,   652,   316,   136,    52,  19,  7,  1,  1]
		

Crossrefs

T(n, 0) = A000957 (Fine), |A002420| (row sums), A000007 (alternating row sums).

Cf. A321620.

Programs

Maple

# The function RiordanSquare is defined in A321620.
Fine := 1 + (1 - sqrt(1 - 4*x))/(3 - sqrt(1 - 4*x)); RiordanSquare(Fine, 10);

Mathematica

(* The function RiordanSquare is defined in A321620. *)
FineGF = 1 + (1 - Sqrt[1 - 4x])/(3 - Sqrt[1 - 4x]);
RiordanSquare[FineGF, 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)

Sage

# uses[riordan_square from A321620]
riordan_square(1 + (1 - sqrt(1 - 4*x))/(3 - sqrt(1 - 4*x)), 10)