A321625 The Riordan square of the swinging factorial (A056040), triangle read by rows, T(n, k) for 0 <= k<= n.

1, 1, 1, 2, 3, 1, 6, 10, 5, 1, 6, 22, 22, 7, 1, 30, 66, 66, 38, 9, 1, 20, 140, 218, 146, 58, 11, 1, 140, 372, 574, 542, 270, 82, 13, 1, 70, 826, 1680, 1708, 1134, 446, 110, 15, 1, 630, 1930, 4156, 5432, 4126, 2106, 682, 142, 17, 1

Offset: 0

Author

Peter Luschny, Nov 22 2018

Keywords

• nonn
• tabl

Examples

			[0] [   1]
[1] [   1,    1]
[2] [   2,    3,    1]
[3] [   6,   10,    5,    1]
[4] [   6,   22,   22,    7,    1]
[5] [  30,   66,   66,   38,    9,    1]
[6] [  20,  140,  218,  146,   58,   11,    1]
[7] [ 140,  372,  574,  542,  270,   82,   13,   1]
[8] [  70,  826, 1680, 1708, 1134,  446,  110,  15,  1]
[9] [ 630, 1930, 4156, 5432, 4126, 2106,  682, 142, 17,  1]
		

Crossrefs

T(n, 0) = A056040 (swinging factorial), A321626 (row sums), A000007 (alternating row sums).

Cf. A321620.

Programs

Maple

# The function RiordanSquare is defined in A321620.
SwingingFactorial := (1 + x/(1 - 4*x^2))/sqrt(1 - 4*x^2);
RiordanSquare(SwingingFactorial, 10);

Mathematica

(* The function RiordanSquare is defined in A321620. *)
SwingingFactorial = (1 + x/(1 - 4*x^2))/Sqrt[1 - 4*x^2];
RiordanSquare[SwingingFactorial, 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)

Sage

# uses[riordan_square from A321620]
riordan_square((1 + x/(1 - 4*x^2))/sqrt(1 - 4*x^2), 10)