OFFSET
0,4
COMMENTS
FORMULA
Since T + T^-1 = C + C^-1, then [T^-1](n,k) = (1+(-1)^(n-k))*C(n,k) - T(n,k) is a formula for the matrix inverse T^-1 = A118435.
EXAMPLE
Triangle begins:
1;
-1, 1;
5,-2, 1;
11,-9,-3, 1;
-23, 44, 30,-4, 1;
-41, 125, 110,-30,-5, 1;
45,-246,-345, 220, 75,-6, 1;
-29,-301,-861, 875, 385,-63,-7, 1;
337,-232, 1260,-2296,-1610, 616, 140,-8, 1;
1199,-3015,-1044,-3612,-5166, 3150, 924,-108,-9, 1; ...
MATHEMATICA
nmax = 12;
h[n_, k_] := Binomial[n, k]*(-1)^(Quotient[n+1, 2] - Quotient[k, 2]+n-k);
H = Table[h[n, k], {n, 0, nmax}, {k, 0, nmax}];
Cn = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, nmax}];
Tn = H.Cn.H;
T[n_, k_] := Tn[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 08 2024 *)
PROG
(PARI) {T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r>=c, binomial(r-1, c-1)*(-1)^(r\2- (c-1)\2+r-c))), C=matrix(n+1, n+1, r, c, if(r>=c, binomial(r-1, c-1)))); (M*C*M)[n+1, k+1]}
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Apr 28 2006
STATUS
approved