This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A107667 #22 Nov 06 2024 04:57:15
%S A107667 1,4,2,45,9,3,816,112,16,4,20225,2200,225,25,5,632700,58176,4860,396,
%T A107667 36,6,23836540,1920163,138817,9408,637,49,7,1048592640,75683648,
%U A107667 4886464,290816,16576,960,64,8,52696514169,3460349970,203451912,10948203,553473
%N 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, ...}.
%F A107667 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.
%F A107667 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]
%e A107667 Reverse of rows form the initial terms of g.f.s below.
%e A107667 Row n=0: 1 = 1*(1-x) + 1*x*(1-x) + ...
%e A107667 Row n=1: 2 = 2*(1-2*x) + 4*x*(1-2*x)*(1-x) + 12*x^2*(1-2*x)*(1-x) + ...
%e A107667 Row n=2: 3 = 3*(1-3*x) + 9*x*(1-3*x)*(1-2*x)
%e A107667 + 45*x^2*(1-3*x)*(1-2*x)*(1-x)
%e A107667 + 216*x^3*(1-3*x)*(1-2*x)*(1-x) + ...
%e A107667 Row n=3: 4 = 4*(1-4*x) + 16*x*(1-4*x)*(1-3*x)
%e A107667 + 112*x^2*(1-4*x)*(1-3*x)*(1-2*x)
%e A107667 + 816*x^3*(1-4*x)*(1-3*x)*(1-2*x)*(1-x)
%e A107667 + 5248*x^4*(1-4*x)*(1-3*x)*(1-2*x)*(1-x) + ...
%e A107667 Triangle T begins:
%e A107667 1;
%e A107667 4, 2;
%e A107667 45, 9, 3;
%e A107667 816, 112, 16, 4;
%e A107667 20225, 2200, 225, 25, 5;
%e A107667 632700, 58176, 4860, 396, 36, 6;
%e A107667 23836540, 1920163, 138817, 9408, 637, 49, 7;
%e A107667 1048592640, 75683648, 4886464, 290816, 16576, 960, 64, 8;
%e A107667 ...
%e A107667 The matrix square T^2 shifts each row right 1 place, dropping the diagonal D and putting A006689 in column 0:
%e A107667 1;
%e A107667 12, 4;
%e A107667 216, 45, 9;
%e A107667 5248, 816, 112, 16;
%e A107667 160675, 20225, 2200, 225, 25;
%e A107667 5931540, 632700, 58176, 4860, 396, 36;
%e A107667 256182290, 23836540, 1920163, 138817, 9408, 637, 49;
%e A107667 ...
%o A107667 (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])}
%o A107667 (Haskell) a = [[sum [a!!n!!i * a!!i!!(k+1) | i<-[k+1..n]] | k <- [0..n-1]] ++ [fromIntegral n+1] | n <- [0..]]
%Y A107667 Cf. A006689, A107668 (column 0), A107669, A107670 (matrix square).
%K A107667 nonn,tabl
%O A107667 0,2
%A A107667 _Paul D. Hanna_, Jun 07 2005