A112292 An invertible triangle of ratios of double factorials.

1, 1, 1, 3, 3, 1, 15, 15, 5, 1, 105, 105, 35, 7, 1, 945, 945, 315, 63, 9, 1, 10395, 10395, 3465, 693, 99, 11, 1, 135135, 135135, 45045, 9009, 1287, 143, 13, 1, 2027025, 2027025, 675675, 135135, 19305, 2145, 195, 15, 1, 34459425, 34459425, 11486475, 2297295, 328185, 36465, 3315, 255, 17, 1

Offset: 0

Author

Paul Barry, Sep 01 2005

Keywords

• easy
• nonn
• tabl

Comments

As a square array read by antidiagonals, column k has e.g.f. (1/(1-2x)^(1/2))*(1/(1-2x))^k. - Paul Barry, Sep 04 2005

Let G(m, k, p) = (-p)^k*Product_{j=0..k-1}(j - m - 1/p) and T(n, k, p) = G(n-1, n-k, p) then T(n, k, 1) = A094587(n, k), T(n, k, 2) is this sequence and T(n, k, 3) = A136214. - Peter Luschny, Jun 01 2009, revised Jun 18 2019

Examples

			Triangle begins
      1;
      1,     1;
      3,     3,    1;
     15,    15,    5,  1;
    105,   105,   35,  7,  1;
    945,   945,  315, 63,  9,  1;
  10395, 10395, 3465,693, 99, 11, 1;
Inverse is A112295, which begins
   1;
  -1,  1;
   0, -3,  1;
   0,  0, -5,  1;
   0,  0,  0, -7,  1;
   0,  0,  0,  0, -9,  1;
Similar results arise for higher factorials.
		

Crossrefs

Columns include A001147, A051577, A051579.

Row sums are A112293.

Diagonal sums are A112294.

Cf. A094587 (p=1), this sequence (p=2), A136214 (p=3).

Programs

Mathematica

T[n_, k_] := If[k <= n, (2n-1)!!/(2k-1)!!, 0];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

Formula

T(n, k)=if(k<=n, (2n-1)!!/(2k-1)!!, 0);

T(n, k)=if(k<=n, n!*C(2n, n)2^(k-n)/(k!*C(2k, k)), 0);

T(n, k)=if(k<=n, 2^(n-k)(n-1/2)!/(k-1/2)!, 0);

T(n, k)=if(k<=n, (n+1)!*C(n)2^(k-n)/((k+1)!*C(k)), 0).