0,2

Matrix diagonalization method: define triangular matrix P by P(n, k) = ((n+1)^2)^(n-k)/(n-k)!, n>=k>=0, and diagonal matrix D(n, n) = n+1, then T is given by T = P^-1*D*P. Also, rows in reverse form the initial terms of the g.f.: (n+1) = Sum_{k>=0} T(n, n-k) * Product_{j=0..k} (1-(n+1-j)*x) = T(n, n)*(1-(n+1)*x) + T(n, n-1)*(1-(n+1)*x)*(1-n*x) + T(n, n-2)*(1-(n+1)*x)*(1-n*x)*(1-(n-1)*x) + ...

Reverse of rows form the initial terms of g.f.s below.

Row 1: 1 = 1*(1-x) + 1*x*(1-x) + ...

Row 2: 2 = 2*(1-2*x) + 4*x*(1-2*x)*(1-x) + 12*x^2*(1-2*x)*(1-x)+...

Row 3: 3 = 3*(1-3*x) + 9*x*(1-3*x)*(1-2*x)

+ 45*x^2*(1-3*x)*(1-2*x)*(1-x)

+ 216*x^3*(1-3*x)*(1-2*x)*(1-x) +...

Row 4: 4 = 4*(1-4*x) + 16*x*(1-4*x)*(1-3*x)

+ 112*x^2*(1-4*x)*(1-3*x)*(1-2*x)

+ 816*x^3*(1-4*x)*(1-3*x)*(1-2*x)*(1-x)

+ 5248*x^4*(1-4*x)*(1-3*x)*(1-2*x)*(1-x) +...

Triangle begins:

1;

4,2;

45,9,3;

816,112,16,4;

20225,2200,225,25,5;

632700,58176,4860,396,36,6;

23836540,1920163,138817,9408,637,49,7;

1048592640,75683648,4886464,290816,16576,960,64,8; ...

The matrix square T^2 shifts each row right 1 place,

dropping the diagonal D and putting A082165 in column 0:

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; ...

(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*P)[n+1, k+1])}

Cf. A107668 (column 0), A107669, A107670 (matrix square), A082165.

Sequence in context: A102015 A123850 A120968 this_sequence A010319 A057167 A096683

Adjacent sequences: A107664 A107665 A107666 this_sequence A107668 A107669 A107670

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 07 2005