A135049 Triangle read by rows: row n gives coefficients of increasing powers of x in the polynomial (-1)^n*p(n), where p(n) is defined as follows. Let f(n) = n*(n+1)/2, g(n) = f(n)+1; then p(-1) = 0, p(0) = 1 and for n >= 1, p(n) = (x - f(n))*p(n - 1) - g(n - 1)^2*p(n - 2).

1, 1, -1, -1, -4, 1, -22, -7, 10, -1, -171, 148, 58, -20, 1, 97, 3238, -488, -237, 35, -1, 45813, 30013, -28334, 631, 716, -56, 1, 1235816, -772641, -587173, 160710, 2477, -1800, 84, -1, 5960643, -54291825, 3463307, 5842062, -673694, -20181, 3983, -120, 1

Offset

1,5

Comments

Inspired by the Cornelius-Schultz article.

References

Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows

E. F. Cornelius Jr. and P. Schultz, Sequences generated by polynomials, Amer. Math. Monthly, No. 2, 2008.

Example

{1},
{1, -1},
{-1, -4, 1},
{-22, -7, 10, -1},
{-171, 148, 58, -20,1},
{97, 3238, -488, -237, 35, -1},
{45813, 30013, -28334, 631, 716, -56, 1},
{1235816, -772641, -587173, 160710, 2477, -1800, 84, -1}.

Mathematica

a[n_] := n*(n + 1)/2; b[n_] = a[n] + 1;
T[n_, m_, d_] := If[ n == m, a[n], If[n == m - 1 || n == m + 1, If[n == m - 1, b[m - 1], If[ n == m + 1, b[n - 1], 0]], 0]]; M0[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; TableForm[Table[M0[n], {n, 1, 4}]];
p1 = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[M0[n + 1], x], x], {n, 0, 10}]]; (* sequence values *)
Flatten[p1] p[x, 0] = 1; p[x, -1] = 0; p[x_, j_] := p[x, j] = (x - a[j])*p[x, j - 1] - b[j - 1]^2*p[x, j - 2]; p2 = Join[{{1}}, Table[CoefficientList[(-1)^n*p[x, n], x], {n, 1, 11}]]; p1 - p2

Crossrefs

Sequence in context: A142472 A360089 A299445 * A113384 A243663 A039812

Adjacent sequences: A135046 A135047 A135048 * A135050 A135051 A135052

Keyword

tabl,sign

Author

Roger L. Bagula, Feb 11 2008

Extensions

Edited by N. J. A. Sloane, Mar 02 2008

Status

approved