Real-rooted atlas

A358735 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(n - n^{2}\right)\,P_{n-4}(x) + \left(-6\,n + 4\,n x + 4\,n^{2} - 6\,n^{2} x + 2\,n^{3} x\right)\,P_{n-3}(x) + \left(-2 + 12\,n - 8\,n x + 2\,n x^{2} - 6\,n^{2} + 12\,n^{2} x - 5\,n^{2} x^{2} - 4\,n^{3} x + 4\,n^{3} x^{2} - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(4 - 10\,n + 4\,n x + 4\,n^{2} - 6\,n^{2} x + 2\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 5 + 8\,x + 2\,x^{2}\)
\(P_{3}(x) = 11 + 42\,x + 34\,x^{2} + 6\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
258215
3114234693
42116029817224675
5364831712220810281205587
657123273681749217848712872051845
785277225764100172183768158728563765040532705

All 15 rows

Technical details
Node id
n-b0fa9c2f4eae3fec · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6b585dfe392e21e2

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   2,     1;
  2:   5,     8,      2;
  3:  11,    42,     34,       6;
  4:  21,   160,    298,     172,      24;
  5:  36,   483,   1712,    2208,    1028,      120;
  6:  57,  1232,   7368,   17492,   17848,     7128,      720;
  7:  85,  2772,  25764,  100172,  183768,   158728,    56376,     5040;
  8: 121,  5664,  77088,  454084, 1335128,  2037688,  1550624,   501408,   40320;
  9: 166, 10725, 204336, 1724736, 7539196, 18142840, 24072256, 16563024, 4956192, 362880;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (4*n^2-6*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (6*n^2-12*n+2+(4*n^3-12*n^2+8*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (4*n^2-10*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 5 + 8*x + 2*x^2, P(3,x) = 11 + 42*x + 34*x^2 + 6*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 5 + 8*x + 2*x^2;
P[3, x_] := 11 + 42*x + 34*x^2 + 6*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 5 + 8*x + 2*x^2; v[4] = 11 + 42*x + 34*x^2 + 6*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*n + 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023