A193229 A double factorial triangle.

1, 1, 1, 3, 3, 2, 15, 15, 12, 6, 105, 105, 90, 60, 24, 945, 945, 840, 630, 360, 120, 10395, 10395, 9450, 7560, 5040, 2520, 720, 135135, 135135, 124740, 103950, 75600, 45360, 20160, 5040, 2027025, 2027025, 1891890, 1621620, 1247400, 831600, 453600, 181440, 40320

Offset: 0

Author

Gary W. Adamson, Jul 18 2011

Keywords

• nonn
• tabl

Comments

The double factorial triangle coefficients are T(n,k), n >= 0 and 0 <= k <= n.

The T(n,0) equal the double factorial numbers A001147(n) = (2*n-1)!!.

The T(n,n) equal the factorial numbers A000142(n) = n!.

The row sums equal the double factorial numbers A000165(n) = (2*n)!!.

The Kn21(n) sums, see A180662 for the definition of these and other triangle sums, equal A130905(n) while the Kn2p(n) sums equal A130905(n+2*p-2) - (n+2*p-2)!*A010844(p-2)/A000165(p-2), p >= 2. - Johannes W. Meijer, Jul 21 2011

Examples

			The first few rows of matrix M[i,j] are:
  1, 1, 0, 0, 0, 0, ...
  2, 2, 2, 0, 0, 0, ...
  3, 3, 3, 3, 0, 0, ...
  4, 4, 4, 4, 4, 0, ...
  5, 5, 5, 5, 5, 5, ...
The first few rows of triangle T(n,k) are:
       1;
       1,      1;
       3,      3,      2;
      15,     15,     12,      6;
     105,    105,     90,     60,    24;
     945,    945,    840,    630,   360,   120;
   10395,  10395,   9450,   7560,  5040,  2520,   720;
  135135, 135135, 124740, 103950, 75600, 45360, 20160, 5040;
		

Links

• Seiichi Manyama, Rows n = 0..139, flattened

Crossrefs

Cf. A000142, A000165, A001147, A102625.

Cf. A180662, A130905, A010844.

T(2*n,n) gives A166334.

Programs

Maple

nmax:=7: M := Matrix(1..nmax+1,1..nmax+1): for i from 1 to nmax do for j from 1 to i+1 do M[i,j] := i od: od: for n from 0 to nmax do B := M^n: for k from 0 to n do T(n,k) := B[1,k+1] 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 21 2011

PARI

row(n)=(matrix(n,n,i,j,(i>j-2)*i)^(n-1))[1,]  \\ M. F. Hasler, Jul 24 2011

Formula

T(n,k) = the (k+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = i, i >= 1 and 1 <= j <= i+1, and M[i,j] = 0, i >= 1 and j >= i+2, see the examples.

It appears that T(n,k) = (2*n-k)!/(2^(n-k)*(n-k)!) with conjectural e.g.f. 1/(x*(1-2*z) + (1-x)*sqrt(1-2*z)) = 1 + (1+x)*z + (3+3*x+2*x^2)*z^2/2! + .... Cf. A102625. - Peter Bala, Jul 09 2012

Extensions

Corrected, edited and extended by Johannes W. Meijer, Jul 21 2011

More terms from Seiichi Manyama, Apr 06 2019