Real-rooted atlas

A249139 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(21 + 19\,x + 4\,x^{2} + 33\,n + 23\,n x + 4\,n x^{2} + 13\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(21 + 19\,x + 4\,x^{2} + 226\,n + 225\,n x + 74\,n x^{2} + 8\,n x^{3} + 86\,n^{2} + 52\,n^{2} x + 8\,n^{2} x^{2} + 7\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-1794 - 2203\,x - 1015\,x^{2} - 208\,x^{3} - 16\,x^{4} - 546\,n - 506\,n x - 156\,n x^{2} - 16\,n x^{3} - 39\,n^{2} - 25\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
1516
29211
32811140
46228393
516598181282
6391266584719
79837932252612028

All 15 rows

Technical details
Node id
n-605c85e110521b73 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-fa00a7f689750668

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    1;
  1:    5,    1;
  2:    9,    2;
  3:   28,   11,    1;
  4:   62,   28,    3;
  5:  165,   98,   18,   1;
  6:  391,  266,   58,   4;
  7:  983,  793,  225,  26,  1;
  8: 2373, 2156,  711, 100,  5;
  9: 5830, 5993, 2338, 425, 35, 1;
%F Formula
The row polynomials satisfy (n^3+13*n^2+33*n+21+(4*n^2+23*n+19)*x+(4*n+4)*x^2)*P(n,x) = (7*n^3+86*n^2+226*n+21+(2*n^3+52*n^2+225*n+19)*x+(8*n^2+74*n+4)*x^2+8*n*x^3)*P(n-1,x) - (39*n^2+546*n+1794+(25*n^2+506*n+2203)*x+(4*n^2+156*n+1015)*x^2+(16*n+208)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((7*n^3 + 86*n^2 + 226*n + 21 + (2*n^3 + 52*n^2 + 225*n + 19)*x + (8*n^2 + 74*n + 4)*x^2 + 8*n*x^3)*P[n - 1, x] - (39*n^2 + 546*n + 1794 + (25*n^2 + 506*n + 2203)*x + (4*n^2 + 156*n + 1015)*x^2 + (16*n + 208)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 + 13*n^2 + 33*n + 21 + (4*n^2 + 23*n + 19)*x + (4*n + 4)*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; for(n = 1, N, my(i = n + 1); v[i] = ((7*n^3 + 86*n^2 + 226*n + 21 + (2*n^3 + 52*n^2 + 225*n + 19)*x + (8*n^2 + 74*n + 4)*x^2 + 8*n*x^3)*v[i - 1] - (39*n^2 + 546*n + 1794 + (25*n^2 + 506*n + 2203)*x + (4*n^2 + 156*n + 1015)*x^2 + (16*n + 208)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 + 13*n^2 + 33*n + 21 + (4*n^2 + 23*n + 19)*x + (4*n + 4)*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. [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.