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.
1, 1, 1, -1, 1, 1, 5, -2, 1, 1, -43, 12, -3, 1, 1, 527, -118, 22, -4, 1, 1, -8396, 1605, -250, 35, -5, 1, 1, 164672, -27816, 3810, -455, 51, -6, 1, 1, -3835910, 585046, -72492, 7735, -749, 70, -7, 1, 1, 103464895, -14459138, 1649634, -161336, 14098, -1148, 92, -8, 1, 1
Offset: 0
Examples
Triangle begins: 1; 1, 1; -1, 1, 1; 5, -2, 1, 1; -43, 12, -3, 1, 1; 527, -118, 22, -4, 1, 1; -8396, 1605, -250, 35, -5, 1, 1; 164672, -27816, 3810, -455, 51, -6, 1, 1; -3835910, 585046, -72492, 7735, -749, 70, -7, 1, 1; 103464895, -14459138, 1649634, -161336, 14098, -1148, 92, -8, 1, 1; ... Matrix inverse T^-1 is a signed version of triangle A101479: 1; -1, 1; 2, -1, 1; -9, 3, -1, 1; 70, -18, 4, -1, 1; -795, 170, -30, 5, -1, 1; 11961, -2220, 335, -45, 6, -1, 1; ... Matrix inverse square T^-2 begins: 1; -2, 1; 5, -2, 1; <-- row 3 of T -23, 7, -2, 1; 175, -43, 9, -2, 1; ... where row 3 of T = row 2 of T^-2 with appended '1'. Matrix inverse cube T^-3 begins: 1; -3, 1; 9, -3, 1; -43, 12, -3, 1; <-- row 4 of T 324, -76, 15, -3, 1; ... where row 4 of T = row 3 of T^-3 with appended '1'. Matrix inverse 4th power T^-4 begins: 1; -4, 1; 14, -4, 1; -70, 18, -4, 1; 527, -118, 22, -4, 1; <-- row 4 of T -5624, 1107, -178, 26, -4, 1; ... where row 5 of T = row 4 of T^-4 with appended '1'.
Programs
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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]))}
Formula
The matrix inverse T^-1 equals triangle A101479 (signed).
Comments