1,8

This kind of tridiagonal while not symmetrical as is traditional in this area

of matrix/ differential equation theory has the benefit of unique inverse matrices that are all Integer. The matrix type was suggested to me by Frank Cornelius in a private email.

This approach seems to give a new type of Pascal like polynomials:

1 - x

1 - x + x^2,

-(-1+x)^3,

(-1+x)^2(1 - 3 x + x^2),

-(-1+x)^3(1 - 4 x + x^2),

(-1+x)^4(1 - 5 x + x^2)

Sequence function: a0(n)=-(n+1) diagonal=a(n,d)= If[n == 1 && d > 1, -1, If[n == 2, -(a0(d) + 1), 1]]; upper sub-diagonal=c(n,d)=If[n == 1, a0(d), 0] lower sub-diagonal=b(n)=1 Matrix: M0(n)=If[ n == m, a(n, d), If[n == m - 1 || n == m + 1, If[n == m - 1, b(m - 1), If[n == m + 1, c(n - 1, d), 0]], 0]

{-1},

{1, -1},

{1, -1, 1},

{1, -3, 3, -1},

{1, -5, 8, -5, 1},

{1, -7, 16, -16, 7, -1},

{1, -9,27, -38, 27, -9, 1},

{1, -11, 41, -75, 75, -41, 11, -1},

{1, -13, 58, -131, 170, -131, 58, -13, 1},

{1, -15, 78, -210, 336, -336, 210, -78, 15, -1},

{1, -17, 101,-316, 602, -742, 602, -316, 101, -17, 1}

Clear[M, T, M0, a, b, p, a0, c, p1, aa] a0[n_] := -(n + 1); a[n_, d_] := If[n == 1 && d > 1, -1, If[n == 2, -(a0[d] + 1), 1]]; c[n_, d_] := If[n == 1, a0[d], 0]; b[n_] := 1; T[n_, m_, d_] := If[ n == m, a[n, d], If[n == m - 1 || n == m + 1, If[n == m - 1, b[m - 1], If[n == m + 1, c[n - 1, d], 0]], 0]]; M0[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; TableForm[Table[M0[n], {n, 1, 4}]]; TableForm[Table[Inverse[M0[n]], {n, 1, 4}]]; p1 = Table[Factor[CharacteristicPolynomial[M0[n], x]], {n, 1, 10}] aa = Join[{{a0[0]}}, Table[CoefficientList[CharacteristicPolynomial[M0[n], x], x], {n, 1, 10}]]; Flatten[aa]

Sequence in context: A046218 A046221 A056611 this_sequence A132729 A034871 A015109

Adjacent sequences: A135666 A135667 A135668 this_sequence A135670 A135671 A135672

nonn,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 16 2008