OFFSET
0,5
COMMENTS
Row sums are: {1,0,-1,0, 1,0,-1,0, ...}. Absolute row sums form A038754. 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)=1+x*y*A(-1/y,-x*y^2) and A(x,y)=(B(-x^2*y,-1/x)-1)/(x*y).
FORMULA
G.f.: A(x, y) = (1 - x + x*y)/(1 + 2*x^2*y - x^2*y^2).
Conjectures from Peter Bala, May 25 2023: (Start)
T(2*n+1,k) = Sum_{i = k-n-1..n} Stirling2(n,i)*Stirling1(i+2,k+1-n) for 0 <= k <= 2*n+1.
T(2*n,k) = binomial(n,k-n)*(-2)^(2*n-k) for 0 <= k <= 2*n. Cf. A038207. (End)
EXAMPLE
Rows of T begin:
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 matrix inverse T^-1 equals triangle A104040:
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; ...
The rows of T^-1 equal columns of T in absolute value.
PROG
(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^2*Y-X^2*Y^2), n, x), k, y)}
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Mar 02 2005
STATUS
approved