Real-rooted atlas

A099093 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(3 + 3\,x - 4\,n - 3\,n x + n^{2}\right)\,P_{n}(x) = \left(-45 - 57\,x - 18\,x^{2} + 23\,n + 15\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-28 - 61\,x - 42\,x^{2} - 9\,x^{3} + 7\,n + 10\,n x + 3\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2211839
311719881396
4585147612153243600
527549234114216156121530780
612474521648610369984279454374253044
754999275562568863618516373005118098153092024352

All 15 rows

Technical details
Node id
n-28f5ed694b21c3a9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a55ee99e24b39585

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 -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:       0;
  1:       3;
  2:      21,      18;
  3:     117,     198,       81;
  4:     585,    1476,     1215,      324;
  5:    2754,    9234,    11421,     6156,     1215;
  6:   12474,   52164,    86103,    69984,    27945,     4374;
  7:   54999,  275562,   568863,   618516,   373005,   118098,   15309;
  8:  237735, 1388016,  3439665,  4685364,  3784725,  1810836,  474579,   52488;
  9: 1012095, 6747138, 19511685, 31945752, 32362740, 20754630, 8220933, 1837080, 177147;
%F Formula
The row polynomials satisfy (n^2-4*n+3-(3*n-3)*x)*P(n,x) = -(2*n^2-23*n+45-(15*n-57)*x+18*x^2)*P(n-1,x) + (7*n-28+(10*n-61)*x+(3*n-42)*x^2-9*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 23*n + 45 - (15*n - 57)*x + 18*x^2)*P[n - 1, x] + (7*n - 28 + (10*n - 61)*x + (3*n - 42)*x^2 - 9*x^3)*D[P[n - 1, x], x])/(n^2 - 4*n + 3 - (3*n - 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 23*n + 45 - (15*n - 57)*x + 18*x^2)*v[i - 1] + (7*n - 28 + (10*n - 61)*x + (3*n - 42)*x^2 - 9*x^3)*deriv(v[i - 1], x))/(n^2 - 4*n + 3 - (3*n - 3)*x)); 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.