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A135049
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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).
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0
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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, -1423603169, -1391347239
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Inspired by the Cornelius-Schultz article.
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REFERENCES
| Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.
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LINKS
| E. F. Cornelius Jr. and P. Schultz, Sequences generated by polynomials, Amer. Math. Monthly, No. 2, 2008.
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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}
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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}]];
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
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CROSSREFS
| Sequence in context: A159841 A202550 A142472 * A113384 A039812 A057869
Adjacent sequences: A135046 A135047 A135048 * A135050 A135051 A135052
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KEYWORD
| tabl,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 11 2008
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Mar 02 2008
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