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%I #17 Jun 16 2020 18:17:14
%S 1,1,0,1,-1,-1,10,-29,-6,19,57,-325,287,423,-19,12813,-120862,291323,
%T 44088,-355855,-227362,1286795,-18146731,79841909,-85635661,
%U -123338281,64989065,145991969,13131073916,-258931801371,1776194531596,-4499161007143,489428412300,8437850634901
%N Polynomials interpolating their own integral coefficients, read by row. The leading coefficients are positive and minimal.
%H Robert Israel, <a href="/A103423/b103423.txt">Table of n, a(n) for n = 0..945</a> (rows 0 to 42, flattened)
%F a(n, k) = Sum_{i=0..n} a(n, i)*k^i, 0<=k<=n.
%e 1;
%e x;
%e x^2-x-1;
%e 10*x^3-29*x^2-6*x+19;
%e 57*x^4-325*x^3+287*x^2+423*x-19;
%e 12813*x^5-120862*x^4+291323*x^3+44088*x^2-355855*x-227362.
%p f:= proc(n) uses LinearAlgebra:
%p local V, d,i;
%p V:= op(NullSpace(VandermondeMatrix([$0..n])-IdentityMatrix(n+1)));
%p if V[-1] < 0 then V:= -V fi;
%p d:= ilcm(seq(denom(V[i]),i=1..n+1));
%p seq(d*V[-i],i=1..n+1)
%p end proc:
%p seq(f(i),i=0..10); # _Robert Israel_, Jan 03 2016
%t VM[n_] := Table[If[k == 1, 1, (j-1)^(k-1)], {j, 1, n}, {k, 1, n}];
%t 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];
%t f /@ Range[0, 10] // Flatten (* _Jean-François Alcover_, Jun 16 2020, after _Robert Israel_ *)
%o (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)
%Y Cf. A103417, A103418.
%K sign,tabl
%O 0,7
%A Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Feb 05 2005
%E More terms from _Max Alekseyev_, Feb 09 2005
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