Real-rooted atlas

A113953 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x - 3\,n - 3\,n x + 2\,n^{2}\right)\,P_{n}(x) = \left(-40 - 54\,x - 18\,x^{2} + 42\,n + 30\,n x - 8\,n^{2}\right)\,P_{n-1}(x) + \left(-9 - 39\,x - 39\,x^{2} - 9\,x^{3} + 6\,n + 20\,n x + 6\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2121830
33312681240
4905048103241728
5219169243744212121511712
65285094181442883619440437476416
7122114274650431448281615958310615309485376

All 15 rows

Technical details
Node id
n-6dcb02e4ee83c36a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4820c45b4dc5077d

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:   12,    18;
  3:   33,   126,     81;
  4:   90,   504,    810,     324;
  5:  219,  1692,   4374,    4212,    1215;
  6:  528,  5094,  18144,   28836,   19440,    4374;
  7: 1221, 14274,  65043,  144828,  161595,   83106,   15309;
  8: 2790, 38124, 211410,  607824,  959850,  813564,  336798,   52488;
  9: 6255, 98064, 641520, 2263464, 4648590, 5598720, 3796632, 1312200, 177147;
%F Formula
The row polynomials satisfy (2*n^2-3*n+1-(3*n-3)*x)*P(n,x) = -(8*n^2-42*n+40-(30*n-54)*x+18*x^2)*P(n-1,x) + (6*n-9+(20*n-39)*x+(6*n-39)*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[(-(8*n^2 - 42*n + 40 - (30*n - 54)*x + 18*x^2)*P[n - 1, x] + (6*n - 9 + (20*n - 39)*x + (6*n - 39)*x^2 - 9*x^3)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1 - (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] = (-(8*n^2 - 42*n + 40 - (30*n - 54)*x + 18*x^2)*v[i - 1] + (6*n - 9 + (20*n - 39)*x + (6*n - 39)*x^2 - 9*x^3)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1 - (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.