A112333 An invertible triangle of ratios of triple factorials.

1, 2, 1, 10, 5, 1, 80, 40, 8, 1, 880, 440, 88, 11, 1, 12320, 6160, 1232, 154, 14, 1, 209440, 104720, 20944, 2618, 238, 17, 1, 4188800, 2094400, 418880, 52360, 4760, 340, 20, 1, 96342400, 48171200, 9634240, 1204280, 109480, 7820, 460, 23, 1, 2504902400

Offset: 0

Author

Paul Barry, Sep 04 2005

Keywords

• easy
• nonn
• tabl

Comments

First column is A008544. Second column is A034000. Third column is A051605. As a square array read by antidiagonals, columns have e.g.f. (1/(1-3x)^(2/3)) * (1/(1-3x))^k.

Examples

			Triangle begins
      1;
      2,    1;
     10,    5,    1;
     80,   40,    8,   1;
    880,  440,   88,  11,  1;
  12320, 6160, 1232, 154, 14, 1;
Inverse triangle A112334 begins
   1;
  -2,  1;
   0, -5,  1;
   0,  0, -8,   1;
   0,  0,  0, -11,   1;
   0,  0,  0,   0, -14,   1;
   0,  0,  0,   0,   0, -17, 1;
		

Programs

Maple

nmax:=8: for n from 0 to nmax do for k from 0 to n do if k<=n then T(n, k) := mul(3*k1-1, k1=1..n)/ mul(3*j-1, j=1..k) else T(n, k) := 0: fi: od: od: for n from 0 to nmax do seq(T(n, k), k=0..n) od: seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jul 04 2011, revised Nov 23 2012

Formula

Number triangle T(n, k)=if(k<=n, Product{k=1..n, 3k-1}/Product{j=1..k, 3j-1}, 0); T(n, k)=if(k<=n, 3^(n-k)*(n-1/3)!/(k-1/3)!, 0).