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A104040 Triangular matrix T, read by rows, such that row n equals the absolute values of column (n+1) in the matrix inverse T^-1 (with extrapolated zeros): T(n,k) = -Sum_{j=1..[n+1/2]} (-1)^j*T(n-j+1,n-2*j+1)*T(n-j,k)) with T(n,n)=1 (n>=0) and T(n,n-1)=n (n>=1). 2

%I

%S 1,1,1,2,2,1,4,4,3,1,8,8,8,4,1,16,16,20,12,5,1,32,32,48,32,18,6,1,64,

%T 64,112,80,56,24,7,1,128,128,256,192,160,80,32,8,1,256,256,576,448,

%U 432,240,120,40,9,1,512,512,1280,1024,1120,672,400,160,50,10,1,1024,1024,2816

%N Triangular matrix T, read by rows, such that row n equals the absolute values of column (n+1) in the matrix inverse T^-1 (with extrapolated zeros): T(n,k) = -Sum_{j=1..[n+1/2]} (-1)^j*T(n-j+1,n-2*j+1)*T(n-j,k)) with T(n,n)=1 (n>=0) and T(n,n-1)=n (n>=1).

%C Row sums are the Pell numbers A000129. Let A(x,y) be the g.f. of T and B(x,y) be the g.f. of T^-1; then B(x,y)=(A(-x^2*y,-1/x)-1)/(x*y) and A(x,y)=1+x*y*B(-1/y,-x*y^2).

%F G.f.: A(x, y) = (1-x+x*y)/(1-2*x-x^2*y^2). T(n, k) = 2*T(n-1, k) + T(n-2, k-2) (n>=k>=2) with T(0, 0)=T(1, 0)=T(1, 1)=1.

%e Rows of T begin:

%e 1;

%e 1,1;

%e 2,2,1;

%e 4,4,3,1;

%e 8,8,8,4,1;

%e 16,16,20,12,5,1;

%e 32,32,48,32,18,6,1;

%e 64,64,112,80,56,24,7,1;

%e 128,128,256,192,160,80,32,8,1; ...

%e The matrix inverse T^-1 equals triangle A104041:

%e 1;

%e -1,1;

%e 0,-2,1;

%e 0,2,-3,1;

%e 0,0,4,-4,1;

%e 0,0,-4,8,-5,1;

%e 0,0,0,-8,12,-6,1;

%e 0,0,0,8,-20,18,-7,1; ...

%e the columns of T^-1 equal rows of T in absolute value.

%o (PARI) T(n,k)=if(n<k || k<0,0,if(n==k,1,if(n==k+1,n, -sum(j=1,(n+1)\2,(-1)^j*T(n-j+1,n-2*j+1)*T(n-j,k)))))

%o (PARI) T(n,k)=if(n<k || k<0,0,if(n==k,1,if(n==k+1,n, 2*T(n-1,k)+if(n>1 <span style="color:#600" title="Replaced deprecated operator & with &&.">&&</span> k>1,T(n-2,k-2)))))

%o (PARI) T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-X+X*Y)/(1-2*X-X^2*Y^2),n,x),k,y)

%Y Cf. A104041, A000129.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Mar 02 2005

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