Real-rooted atlas

A113214 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(3\,n\right)\,P_{n}(x) = \left(-3\,x - 5\,x^{2} + 9\,n + 12\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-9\,x - 18\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
2915428
32763397136
4812432348711656
52438911161675180183168
672931595184413117103542915296
721871093521627219511244739876754773856

All 15 rows

Technical details
Node id
n-d564dc9f20dfade3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-fcdb948664301fc0

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 0)
  • 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:     3,      3;
  2:     9,     15,      4;
  3:    27,     63,     39,      7;
  4:    81,    243,    234,     87,     11;
  5:   243,    891,   1161,    675,    180,     18;
  6:   729,   3159,   5184,   4131,   1710,    354,    29;
  7:  2187,  10935,  21627,  21951,  12447,   3987,   675,    47;
  8:  6561,  37179,  86022, 106191,  76869,  33669,  8784,  1257,   76;
  9: 19683, 124659, 330237, 480411, 424278, 237168, 84429, 18567, 2301, 123;
%F Formula
The row polynomials satisfy 3*n*P(n,x) = (9*n+(12*n-3)*x+(5*n-5)*x^2)*P(n-1,x) - (9*x+18*x^2+5*x^3)*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[((9*n + (12*n - 3)*x + (5*n - 5)*x^2)*P[n - 1, x] - (9*x + 18*x^2 + 5*x^3)*D[P[n - 1, x], x])/(3*n)]];
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] = ((9*n + (12*n - 3)*x + (5*n - 5)*x^2)*v[i - 1] - (9*x + 18*x^2 + 5*x^3)*deriv(v[i - 1], x))/(3*n)); 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

Same fingerprint ?

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