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

0,4

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

Links

Table of n, a(n) for n=0..54.

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).

Example

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.

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 *)

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).

Sequence in context: A216294 A039797 A143171 * A001497 A123244 A364505

Adjacent sequences: A112289 A112290 A112291 * A112293 A112294 A112295

Keyword

easy,nonn,tabl

Author

Paul Barry, Sep 01 2005

Status

approved