OFFSET
0,39
COMMENTS
FORMULA
G.f.: A(x,y) = 1/(1-x*y)+ x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y)/(1-x*y). G.f. of matrix power T^m: 1/(1-x*y)+ m*x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y)/(1-x*y).
EXAMPLE
Matrix inverse is: T^-1 = 2*I - T.
Matrix log is: log(T) = T - I.
Triangle T begins:
1;
1, 1;
1, 0, 1;
1,-1, 1, 1;
1, 0, 0, 0, 1;
1,-1, 0, 0, 1, 1;
1, 0, 1, 0,-1, 0, 1;
1,-1,-1,-1, 1, 1, 1, 1;
1, 0, 2, 2, 0,-2,-2, 0, 1;
1,-1,-2,-4,-2, 2, 4, 2, 1, 1;
1, 0, 3, 6, 6, 0,-6,-6,-3, 0, 1;
1,-1,-3,-9,-12,-6, 6, 12, 9, 3, 1, 1;
1, 0, 4, 12, 21, 18, 0,-18,-21,-12,-4, 0, 1; ...
The g.f. of column k, C_k(x), obeys the recurrence:
C_k = C_{k-1} + (-1)^k*x*(1+2*x)/(1-x)/(1+x)^k with C_0 = 1/(1-x);
so that column g.f.s continue as:
C_1 = C_0 - x*(1+2*x)/(1-x)/(1+x),
C_2 = C_1 + x*(1+2*x)/(1-x)/(1+x)^2,
C_3 = C_2 - x*(1+2*x)/(1-x)/(1+x)^3, ...
PROG
(PARI) T(n, k)=local(x=X+X*O(X^n), y=Y+Y*O(Y^k)); polcoeff(polcoeff( 1/(1-x*y)+ x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y)/(1-x*y), n, X), k, Y)
(PARI) T(n, k)=local(M=matrix(n+1, n+1)); for(r=1, n+1, for(c=1, r, M[r, c]=if(r==c, 1, if(c==1, 1, if(c>1, (2*M^0-M)[r-1, c-1])+(2*M^0-M)[r-1, c])))); return(M[n+1, k+1])
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Feb 19 2006
STATUS
approved