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).

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, 64, 112, 80, 56, 24, 7, 1, 128, 128, 256, 192, 160, 80, 32, 8, 1, 256, 256, 576, 448, 432, 240, 120, 40, 9, 1, 512, 512, 1280, 1024, 1120, 672, 400, 160, 50, 10, 1, 1024, 1024, 2816

Offset

0,4

Comments

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).

Links

Table of n, a(n) for n=0..68.

Formula

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.

Example

Rows of T begin:
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;
64,64,112,80,56,24,7,1;
128,128,256,192,160,80,32,8,1; ...
The matrix inverse T^-1 equals triangle A104041:
1;
-1,1;
0,-2,1;
0,2,-3,1;
0,0,4,-4,1;
0,0,-4,8,-5,1;
0,0,0,-8,12,-6,1;
0,0,0,8,-20,18,-7,1; ...
the columns of T^-1 equal rows of T in absolute value.

Prog

(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)))))
(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 && k>1, T(n-2, k-2)))))
(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)

Crossrefs

Cf. A104041, A000129.

Sequence in context: A124725 A106522 A128175 * A338131 A332601 A348840

Adjacent sequences: A104037 A104038 A104039 * A104041 A104042 A104043

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 02 2005

Status

approved