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A132690 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.

%I A132690 #5 Mar 14 2015 12:04:44
%S A132690 1,1,1,-1,1,1,5,-2,1,1,-43,12,-3,1,1,527,-118,22,-4,1,1,-8396,1605,
%T A132690 -250,35,-5,1,1,164672,-27816,3810,-455,51,-6,1,1,-3835910,585046,
%U A132690 -72492,7735,-749,70,-7,1,1,103464895,-14459138,1649634,-161336,14098,-1148,92,-8,1,1
%N A132690 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 A132690 The matrix inverse T^-1 equals triangle A101479 (signed).
%e A132690 Triangle begins:
%e A132690 1;
%e A132690 1, 1;
%e A132690 -1, 1, 1;
%e A132690 5, -2, 1, 1;
%e A132690 -43, 12, -3, 1, 1;
%e A132690 527, -118, 22, -4, 1, 1;
%e A132690 -8396, 1605, -250, 35, -5, 1, 1;
%e A132690 164672, -27816, 3810, -455, 51, -6, 1, 1;
%e A132690 -3835910, 585046, -72492, 7735, -749, 70, -7, 1, 1;
%e A132690 103464895, -14459138, 1649634, -161336, 14098, -1148, 92, -8, 1, 1; ...
%e A132690 Matrix inverse T^-1 is a signed version of triangle A101479:
%e A132690 1;
%e A132690 -1, 1;
%e A132690 2, -1, 1;
%e A132690 -9, 3, -1, 1;
%e A132690 70, -18, 4, -1, 1;
%e A132690 -795, 170, -30, 5, -1, 1;
%e A132690 11961, -2220, 335, -45, 6, -1, 1; ...
%e A132690 Matrix inverse square T^-2 begins:
%e A132690 1;
%e A132690 -2, 1;
%e A132690 5, -2, 1; <-- row 3 of T
%e A132690 -23, 7, -2, 1;
%e A132690 175, -43, 9, -2, 1; ...
%e A132690 where row 3 of T = row 2 of T^-2 with appended '1'.
%e A132690 Matrix inverse cube T^-3 begins:
%e A132690 1;
%e A132690 -3, 1;
%e A132690 9, -3, 1;
%e A132690 -43, 12, -3, 1; <-- row 4 of T
%e A132690 324, -76, 15, -3, 1; ...
%e A132690 where row 4 of T = row 3 of T^-3 with appended '1'.
%e A132690 Matrix inverse 4th power T^-4 begins:
%e A132690 1;
%e A132690 -4, 1;
%e A132690 14, -4, 1;
%e A132690 -70, 18, -4, 1;
%e A132690 527, -118, 22, -4, 1; <-- row 4 of T
%e A132690 -5624, 1107, -178, 26, -4, 1; ...
%e A132690 where row 5 of T = row 4 of T^-4 with appended '1'.
%o A132690 (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 A132690 Cf. A101479; columns: A132691, A132692, A132693, A132694.
%K A132690 tabl,sign
%O A132690 0,7
%A A132690 _Paul D. Hanna_, Aug 25 2007