login
Matrix square of triangle A107667.
4

%I #11 Mar 11 2021 03:04:15

%S 1,12,4,216,45,9,5248,816,112,16,160675,20225,2200,225,25,5931540,

%T 632700,58176,4860,396,36,256182290,23836540,1920163,138817,9408,637,

%U 49,12665445248,1048592640,75683648,4886464,290816,16576,960,64

%N Matrix square of triangle A107667.

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

%F 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.

%e Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

%e 1;

%e 12, 4;

%e 216, 45, 9;

%e 5248, 816, 112, 16;

%e 160675, 20225, 2200, 225, 25;

%e 5931540, 632700, 58176, 4860, 396, 36;

%e 256182290, 23836540, 1920163, 138817, 9408, 637, 49;

%e ...

%o (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])}

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

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Jun 07 2005