%I #5 Mar 14 2015 12:04:44
%S 1,1,1,-1,1,1,5,-2,1,1,-43,12,-3,1,1,527,-118,22,-4,1,1,-8396,1605,
%T -250,35,-5,1,1,164672,-27816,3810,-455,51,-6,1,1,-3835910,585046,
%U -72492,7735,-749,70,-7,1,1,103464895,-14459138,1649634,-161336,14098,-1148,92,-8,1,1
%N Triangle T, read by rows, where row n+1 of T = row n of T^(-n) with appended '1' for n>=0 with T(0,0)=1.
%F The matrix inverse T^-1 equals triangle A101479 (signed).
%e Triangle begins:
%e 1;
%e 1, 1;
%e -1, 1, 1;
%e 5, -2, 1, 1;
%e -43, 12, -3, 1, 1;
%e 527, -118, 22, -4, 1, 1;
%e -8396, 1605, -250, 35, -5, 1, 1;
%e 164672, -27816, 3810, -455, 51, -6, 1, 1;
%e -3835910, 585046, -72492, 7735, -749, 70, -7, 1, 1;
%e 103464895, -14459138, 1649634, -161336, 14098, -1148, 92, -8, 1, 1; ...
%e Matrix inverse T^-1 is a signed version of triangle A101479:
%e 1;
%e -1, 1;
%e 2, -1, 1;
%e -9, 3, -1, 1;
%e 70, -18, 4, -1, 1;
%e -795, 170, -30, 5, -1, 1;
%e 11961, -2220, 335, -45, 6, -1, 1; ...
%e Matrix inverse square T^-2 begins:
%e 1;
%e -2, 1;
%e 5, -2, 1; <-- row 3 of T
%e -23, 7, -2, 1;
%e 175, -43, 9, -2, 1; ...
%e where row 3 of T = row 2 of T^-2 with appended '1'.
%e Matrix inverse cube T^-3 begins:
%e 1;
%e -3, 1;
%e 9, -3, 1;
%e -43, 12, -3, 1; <-- row 4 of T
%e 324, -76, 15, -3, 1; ...
%e where row 4 of T = row 3 of T^-3 with appended '1'.
%e Matrix inverse 4th power T^-4 begins:
%e 1;
%e -4, 1;
%e 14, -4, 1;
%e -70, 18, -4, 1;
%e 527, -118, 22, -4, 1; <-- row 4 of T
%e -5624, 1107, -178, 26, -4, 1; ...
%e where row 5 of T = row 4 of T^-4 with appended '1'.
%o (PARI) {T(n, k)=local(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(A^(-(i-2)))[i-1, j]); )); A=B); return( ((A)[n+1, k+1]))}
%Y Cf. A101479; columns: A132691, A132692, A132693, A132694.
%K tabl,sign
%O 0,7
%A _Paul D. Hanna_, Aug 25 2007