Real-rooted atlas

A063007 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + x + x^{2} + 6\,n - n x - n x^{2} - 4\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(1 + 7\,x + 7\,x^{2} + 2\,x^{3} + 2\,n - 5\,n x - 7\,n x^{2} - 2\,n x^{3} - 6\,n^{2} - 4\,n^{2} x + 3\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-4 - 18\,x - 24\,x^{2} - 12\,x^{3} - 2\,x^{4} - 8\,n - 12\,n x - 4\,n x^{2} + 4\,n^{2} + 6\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2527
33130667
419531215020677
5122127602190700706941
67593222902499013160315025271435
7469151698062473801839607245013860924735295
828826312420802232468215712011988903769926006034327559305

All 15 rows

Technical details
Node id
n-ce137bd5ab0d5fca · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-933bf7e3a34d1923

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        5,        2;
   3:       31,       30,         6;
   4:      195,      312,       150,        20;
   5:     1221,     2760,      2190,       700,        70;
   6:     7593,    22290,     24990,     13160,      3150,       252;
   7:    46915,   169806,    247380,    183960,     72450,     13860,      924;
   8:   288263,  1242080,   2232468,   2157120,   1198890,    376992,    60060,    3432;
   9:  1762825,  8817696,  18878580,  22508640,  16250850,   7207200,  1885884,  257400,   12870;
  10: 10736973, 61178130, 152133300, 216114360, 192522330, 110882772, 40900860, 9163440, 1093950, 48620;
%F Formula
The row polynomials satisfy (n^3-4*n^2+6*n-3-(n-1)*x-(n-1)*x^2)*P(n,x) = (3*n^3-6*n^2+2*n+1+(2*n^3-4*n^2-5*n+7)*x-(7*n-7)*x^2-(2*n-2)*x^3)*P(n-1,x) + (4*n^2-8*n-4+(6*n^2-12*n-18)*x+(2*n^2-4*n-24)*x^2-12*x^3-2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 6*n^2 + 2*n + 1 + (2*n^3 - 4*n^2 - 5*n + 7)*x - (7*n - 7)*x^2 - (2*n - 2)*x^3)*P[n - 1, x] + (4*n^2 - 8*n - 4 + (6*n^2 - 12*n - 18)*x + (2*n^2 - 4*n - 24)*x^2 - 12*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + 6*n - 3 - (n - 1)*x - (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n^3 - 6*n^2 + 2*n + 1 + (2*n^3 - 4*n^2 - 5*n + 7)*x - (7*n - 7)*x^2 - (2*n - 2)*x^3)*v[i - 1] + (4*n^2 - 8*n - 4 + (6*n^2 - 12*n - 18)*x + (2*n^2 - 4*n - 24)*x^2 - 12*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + 6*n - 3 - (n - 1)*x - (n - 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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