Real-rooted atlas

A367270 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x - n x^{2} - 5\,n^{2} x + n^{2} x^{2} - 2\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(-6 - 10\,x - 2\,x^{2} + 2\,x^{3} + 16\,n + 30\,n x + 8\,n x^{2} - 5\,n x^{3} - 8\,n^{2} - 25\,n^{2} x - 12\,n^{2} x^{2} + 3\,n^{2} x^{3} - 6\,n^{3} + 2\,n^{3} x + 6\,n^{3} x^{2} + 4\,n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-2\,x - 3\,x^{2} + x^{4} + 4\,n x + 8\,n x^{2} + 2\,n x^{3} - 2\,n x^{4} - 4\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2527
32218343
495120424261
54067003908051581
617223780294098013569563
77260194041953092402100210757751

All 15 rows

Technical details
Node id
n-d37de8e23475b650 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2f3702dc82e1aa6a

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:      1;
  2:      5,      2;
  3:     22,     18,      3;
  4:     95,    120,     42,      4;
  5:    406,    700,    390,     80,      5;
  6:   1722,   3780,   2940,    980,    135,     6;
  7:   7260,  19404,  19530,   9240,   2100,   210,    7;
  8:  30459,  96096, 119196,  73920,  24150,  4032,  308,   8;
  9: 127270, 463320, 684684, 528528, 228690, 55440, 7140, 432, 9;
%F Formula
The row polynomials satisfy (n^4-2*n^3+n+(2*n^3-5*n^2+3*n)*x+(n^2-n)*x^2)*P(n,x) = (4*n^4-6*n^3-8*n^2+16*n-6+(3*n^4+2*n^3-25*n^2+30*n-10)*x+(6*n^3-12*n^2+8*n-2)*x^2+(3*n^2-5*n+2)*x^3)*P(n-1,x) - ((2*n^3-4*n+2)*x+(2*n^3+4*n^2-8*n+3)*x^2+(4*n^2-2*n)*x^3+(2*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^4 - 6*n^3 - 8*n^2 + 16*n - 6 + (3*n^4 + 2*n^3 - 25*n^2 + 30*n - 10)*x + (6*n^3 - 12*n^2 + 8*n - 2)*x^2 + (3*n^2 - 5*n + 2)*x^3)*P[n - 1, x] - ((2*n^3 - 4*n + 2)*x + (2*n^3 + 4*n^2 - 8*n + 3)*x^2 + (4*n^2 - 2*n)*x^3 + (2*n - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 2*n^3 + n + (2*n^3 - 5*n^2 + 3*n)*x + (n^2 - n)*x^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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^4 - 6*n^3 - 8*n^2 + 16*n - 6 + (3*n^4 + 2*n^3 - 25*n^2 + 30*n - 10)*x + (6*n^3 - 12*n^2 + 8*n - 2)*x^2 + (3*n^2 - 5*n + 2)*x^3)*v[i - 1] - ((2*n^3 - 4*n + 2)*x + (2*n^3 + 4*n^2 - 8*n + 3)*x^2 + (4*n^2 - 2*n)*x^3 + (2*n - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 2*n^3 + n + (2*n^3 - 5*n^2 + 3*n)*x + (n^2 - n)*x^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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023