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

%I A321627 #16 Feb 17 2020 11:37:36
%S A321627 1,1,1,3,4,1,15,21,7,1,105,144,48,10,1,945,1245,372,84,13,1,10395,
%T A321627 13140,3357,726,129,16,1,135135,164745,35415,6873,1233,183,19,1,
%U A321627 2027025,2399040,434520,73116,12306,1920,246,22,1
%N A321627 The Riordan square of the double factorial of odd numbers. Triangle T(n, k), 0 <= k <= n, read by rows.
%C A321627 The Riordan square is defined in A321620.
%C A321627 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
%e A321627 Triangle starts:
%e A321627 [0][     1]
%e A321627 [1][     1,      1]
%e A321627 [2][     3,      4,     1]
%e A321627 [3][    15,     21,     7,    1]
%e A321627 [4][   105,    144,    48,   10,    1]
%e A321627 [5][   945,   1245,   372,   84,   13,   1]
%e A321627 [6][ 10395,  13140,  3357,  726,  129,  16,  1]
%e A321627 [7][135135, 164745, 35415, 6873, 1233, 183, 19, 1]
%p A321627 # The function RiordanSquare is defined in A321620.
%p A321627 cf := proc(dim) local k, m; m := 1;
%p A321627 for k from dim by -1 to 1 do m := 1 - k*x/m od;
%p A321627 1/m end: RiordanSquare(cf(9), 9);
%t A321627 (* The function RiordanSquare is defined in A321620. *)
%t A321627 cf[dim_] := Module[{k, m=1}, For[k=dim, k >= 1, k--, m = 1 - k*x/m]; 1/m];
%t A321627 RiordanSquare[cf[9], 9] (* _Jean-François Alcover_, Jun 15 2019, from Maple *)
%Y A321627 First column are the double factorial of odd numbers A001147.
%Y A321627 Second column is number of singletons in pair-partitions A233481.
%Y A321627 Row sums are A321628, alternating row sums are A000007.
%Y A321627 Cf. A321620.
%K A321627 nonn,tabl
%O A321627 0,4
%A A321627 _Peter Luschny_, Dec 07 2018