A103423 Polynomials interpolating their own integral coefficients, read by row. The leading coefficients are positive and minimal.

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

Offset

0,7

Links

Robert Israel, Table of n, a(n) for n = 0..945 (rows 0 to 42, flattened)

Formula

a(n, k) = Sum_{i=0..n} a(n, i)*k^i, 0<=k<=n.

Example

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.

Maple

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:
seq(f(i), i=0..10); # Robert Israel, Jan 03 2016

Mathematica

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];
f /@ Range[0, 10] // Flatten (* Jean-François Alcover, Jun 16 2020, after Robert Israel *)

Prog

(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)

Crossrefs

Cf. A103417, A103418.

Sequence in context: A345696 A066527 A343385 * A102542 A098751 A143190

Adjacent sequences: A103420 A103421 A103422 * A103424 A103425 A103426

Keyword

sign,tabl

Author

Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Feb 05 2005

Extensions

More terms from Max Alekseyev, Feb 09 2005

Status

approved