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A107670 Matrix square of triangle A107667.

%I A107670 #11 Mar 11 2021 03:04:15
%S A107670 1,12,4,216,45,9,5248,816,112,16,160675,20225,2200,225,25,5931540,
%T A107670 632700,58176,4860,396,36,256182290,23836540,1920163,138817,9408,637,
%U A107670 49,12665445248,1048592640,75683648,4886464,290816,16576,960,64
%N A107670 Matrix square of triangle A107667.
%C A107670 Column 0 is A006689. See triangle A107667 for more formulas.
%F A107670 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 A107670 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A107670           1;
%e A107670          12,        4;
%e A107670         216,       45,       9;
%e A107670        5248,      816,     112,     16;
%e A107670      160675,    20225,    2200,    225,   25;
%e A107670     5931540,   632700,   58176,   4860,  396,  36;
%e A107670   256182290, 23836540, 1920163, 138817, 9408, 637, 49;
%e A107670   ...
%o A107670 (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 A107670 Cf. A107667, A107668, A107669, A006689 (column 0).
%K A107670 nonn,tabl
%O A107670 0,2
%A A107670 _Paul D. Hanna_, Jun 07 2005