A107670 Matrix square of triangle A107667.

1, 12, 4, 216, 45, 9, 5248, 816, 112, 16, 160675, 20225, 2200, 225, 25, 5931540, 632700, 58176, 4860, 396, 36, 256182290, 23836540, 1920163, 138817, 9408, 637, 49, 12665445248, 1048592640, 75683648, 4886464, 290816, 16576, 960, 64

Offset

0,2

Comments

Column 0 is A006689. See triangle A107667 for more formulas.

Links

Table of n, a(n) for n=0..35.

Formula

Matrix diagonalization method: define the triangular matrix P by P(n, k) = ((n+1)^2)^(n-k)/(n-k)! for n >= k >= 0 and the diagonal matrix D by D(n, n) = n+1 for n >= 0; then T is given by T = P^-1*D^2*P.

Example

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
          1;
         12,        4;
        216,       45,       9;
       5248,      816,     112,     16;
     160675,    20225,    2200,    225,   25;
    5931540,   632700,   58176,   4860,  396,  36;
  256182290, 23836540, 1920163, 138817, 9408, 637, 49;
  ...

Prog

(PARI) {T(n, k)=local(P=matrix(n+1, n+1, r, c, if(r>=c, (r^2)^(r-c)/(r-c)!)), D=matrix(n+1, n+1, r, c, if(r==c, r))); if(n>=k, (P^-1*D^2*P)[n+1, k+1])}

Crossrefs

Cf. A107667, A107668, A107669, A006689 (column 0).

Sequence in context: A002911 A038330 A144630 * A157782 A335949 A002679

Adjacent sequences: A107667 A107668 A107669 * A107671 A107672 A107673

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 07 2005

Status

approved