A135049 - OEIS

%I #10 Sep 26 2016 21:29:31

%S 1,1,-1,-1,-4,1,-22,-7,10,-1,-171,148,58,-20,1,97,3238,-488,-237,35,

%T -1,45813,30013,-28334,631,716,-56,1,1235816,-772641,-587173,160710,

%U 2477,-1800,84,-1,5960643,-54291825,3463307,5842062,-673694,-20181,3983,-120,1

%N 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).

%C Inspired by the Cornelius-Schultz article.

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

%H G. C. Greubel, <a href="/A135049/b135049.txt">Table of n, a(n) for the first 50 rows</a>

%H E. F. Cornelius Jr. and P. Schultz, <a href="http://www.jstor.org/stable/27642423">Sequences generated by polynomials</a>, Amer. Math. Monthly, No. 2, 2008.

%e {1},

%e {1, -1},

%e {-1, -4, 1},

%e {-22, -7, 10, -1},

%e {-171, 148, 58, -20,1},

%e {97, 3238, -488, -237, 35, -1},

%e {45813, 30013, -28334, 631, 716, -56, 1},

%e {1235816, -772641, -587173, 160710, 2477, -1800, 84, -1}.

%t a[n_] := n*(n + 1)/2; b[n_] = a[n] + 1;

%t 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}]];

%t p1 = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[M0[n + 1], x], x], {n, 0, 10}]]; (* sequence values *)

%t 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

%K tabl,sign

%O 1,5

%A _Roger L. Bagula_, Feb 11 2008

%E Edited by _N. J. A. Sloane_, Mar 02 2008