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A103423
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Polynomials interpolating their own integral coefficients, read by row. The leading coefficients are positive and minimal.
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3
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1, 1, 0, 1, -1, -1, 10, -29, -6, 19, 57, -325, 287, 423, -19, 12813, -120862, 291323, 44088, -355855, -227362, 1286795, -18146731, 79841909, -85635661, -123338281, 64989065, 145991969, 13131073916, -258931801371, 1776194531596, -4499161007143, 489428412300, 8437850634901
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OFFSET
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0,7
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LINKS
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FORMULA
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a(n, k) = Sum_{i=0..n} a(n, i)*k^i, 0<=k<=n.
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EXAMPLE
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1;
x;
x^2-x-1;
10*x^3-29*x^2-6*x+19;
57*x^4-325*x^3+287*x^2+423*x-19;
12813*x^5-120862*x^4+291323*x^3+44088*x^2-355855*x-227362.
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MAPLE
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f:= proc(n) uses LinearAlgebra:
local V, d, i;
V:= op(NullSpace(VandermondeMatrix([$0..n])-IdentityMatrix(n+1)));
if V[-1] < 0 then V:= -V fi;
d:= ilcm(seq(denom(V[i]), i=1..n+1));
seq(d*V[-i], i=1..n+1)
end proc:
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MATHEMATICA
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VM[n_] := Table[If[k == 1, 1, (j-1)^(k-1)], {j, 1, n}, {k, 1, n}];
f[n_] := Module[{V, d}, V = NullSpace[VM[n+1] - IdentityMatrix[n+1]][[1]]; If[V[[-1]] < 0, V = -V]; d = LCM @@ Denominator[V]; d V // Reverse];
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PROG
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(PARI) { f(n) = local(v); v=matkerint(matrix(n+1, n+1, i, j, (i-1)^(j-1)-(i==j))); c=vector(n+1, i, v[n+2-i, 1]); if(c[1]<0, for(i=1, n+1, c[i]=-c[i])); return(c); } \\ function f(n) generate coefficients of the polynomial of degree n (Alekseyev)
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CROSSREFS
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KEYWORD
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AUTHOR
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Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Feb 05 2005
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EXTENSIONS
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STATUS
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approved
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