A107667 Triangular matrix T, read by rows, that satisfies: T = D + SHIFT_LEFT(T^2) where SHIFT_LEFT shifts each row 1 place to the left and D is the diagonal matrix {1, 2, 3, ...}.

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, 52696514169, 3460349970, 203451912, 10948203, 553473

Offset: 0

Author

Paul D. Hanna, Jun 07 2005

Keywords

• nonn
• tabl

Examples

			Reverse of rows form the initial terms of g.f.s below.
Row n=0: 1 = 1*(1-x) + 1*x*(1-x) + ...
Row n=1: 2 = 2*(1-2*x) + 4*x*(1-2*x)*(1-x) + 12*x^2*(1-2*x)*(1-x) + ...
Row n=2: 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 n=3: 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 T 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 A006689 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;
  ...
		

Crossrefs

Cf. A006689, A107668 (column 0), A107669, A107670 (matrix square).

Programs

Haskell

a = [[sum [a!!n!!i * a!!i!!(k+1) | i<-[k+1..n]] | k <- [0..n-1]] ++ [fromIntegral n+1] | n <- [0..]]

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])}

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*P.

Rows read in reverse form the initial terms of the g.f.: (n+1) = Sum_{k>=0} T(n, n-k) * x^k * Product_{j=0..k} (1-(n+1-j)*x) = T(n, n)*(1-(n+1)*x) + T(n, n-1)*x*(1-(n+1)*x)*(1-n*x) + T(n, n-2)*x^2*(1-(n+1)*x)*(1-n*x)*(1-(n-1)*x) + ... [Corrected by Petros Hadjicostas, Mar 11 2021]