Real-rooted atlas

A063967 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 2\,x - 5\,n - n x + n^{2}\right)\,P_{n}(x) = \left(-48 - 20\,x - 2\,x^{2} + 22\,n + 5\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-48 - 44\,x - 12\,x^{2} - x^{3} + 12\,n + 8\,n x + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3527
42016339
57286334195
6244388225565918
7796158012274648564158
82528601658562976830120718333

All 15 rows

Technical details
Node id
n-b957a61ea35749b6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-cb308d1679a71119

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:     0;
   2:     1;
   3:     5,     2;
   4:    20,    16,      3;
   5:    72,    86,     33,     4;
   6:   244,   388,    225,    56,     5;
   7:   796,  1580,   1227,   464,    85,     6;
   8:  2528,  6016,   5856,  2976,   830,   120,    7;
   9:  7872, 21840,  25560, 16324,  6130,  1350,  161,   8;
  10: 24144, 76512, 104616, 80488, 38045, 11292, 2051, 208, 9;
%F Formula
The row polynomials satisfy (n^2-5*n+6-(n-2)*x)*P(n,x) = -(2*n^2-22*n+48-(5*n-20)*x+2*x^2)*P(n-1,x) + (12*n-48+(8*n-44)*x+(n-12)*x^2-x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 22*n + 48 - (5*n - 20)*x + 2*x^2)*P[n - 1, x] + (12*n - 48 + (8*n - 44)*x + (n - 12)*x^2 - x^3)*D[P[n - 1, x], x])/(n^2 - 5*n + 6 - (n - 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 22*n + 48 - (5*n - 20)*x + 2*x^2)*v[i - 1] + (12*n - 48 + (8*n - 44)*x + (n - 12)*x^2 - x^3)*deriv(v[i - 1], x))/(n^2 - 5*n + 6 - (n - 2)*x)); 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. [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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.