A321627 The Riordan square of the double factorial of odd numbers. Triangle T(n, k), 0 <= k <= n, read by rows.

1, 1, 1, 3, 4, 1, 15, 21, 7, 1, 105, 144, 48, 10, 1, 945, 1245, 372, 84, 13, 1, 10395, 13140, 3357, 726, 129, 16, 1, 135135, 164745, 35415, 6873, 1233, 183, 19, 1, 2027025, 2399040, 434520, 73116, 12306, 1920, 246, 22, 1

Offset: 0

Author

Peter Luschny, Dec 07 2018

Keywords

• nonn
• tabl

Comments

The Riordan square is defined in A321620.

Triangle, read by rows, given by [1, 2, 3, 4, 5, 6, 7, …] DELTA [1, 0, 0, 0, 0, 0, 0, 0, …] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 17 2020

Examples

			Triangle starts:
[0][     1]
[1][     1,      1]
[2][     3,      4,     1]
[3][    15,     21,     7,    1]
[4][   105,    144,    48,   10,    1]
[5][   945,   1245,   372,   84,   13,   1]
[6][ 10395,  13140,  3357,  726,  129,  16,  1]
[7][135135, 164745, 35415, 6873, 1233, 183, 19, 1]
		

Crossrefs

First column are the double factorial of odd numbers A001147.

Second column is number of singletons in pair-partitions A233481.

Row sums are A321628, alternating row sums are A000007.

Cf. A321620.

Programs

Maple

# The function RiordanSquare is defined in A321620.
cf := proc(dim) local k, m; m := 1;
for k from dim by -1 to 1 do m := 1 - k*x/m od;
1/m end: RiordanSquare(cf(9), 9);

Mathematica

(* The function RiordanSquare is defined in A321620. *)
cf[dim_] := Module[{k, m=1}, For[k=dim, k >= 1, k--, m = 1 - k*x/m]; 1/m];
RiordanSquare[cf[9], 9] (* Jean-François Alcover, Jun 15 2019, from Maple *)