Real-rooted atlas

A026729 → basis A008277 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x - 2\,x^{2} + n x^{2}\right)\,P_{n-2}(x) + \left(-x^{3}\right)\,P_{n-2}'(x) + \left(1 - x + n x\right)\,P_{n-1}(x) + \left(-x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
31427
41813325
511344395102
61191082081128456
7126221736907315132219
81344022044449137728752111640

All 15 rows

Technical details
Node id
n-88c78ad5290c6424 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-967381822cbe5980

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  4,    2;
   4: 1,  8,   13,     3;
   5: 1, 13,   44,    39,     5;
   6: 1, 19,  108,   208,   112,      8;
   7: 1, 26,  221,   736,   907,    315,     13;
   8: 1, 34,  402,  2044,  4491,   3772,    875,    21;
   9: 1, 43,  673,  4833, 16552,  25587,  15227,  2414,   34;
  10: 1, 53, 1059, 10184, 50006, 123058, 139311, 60286, 6633, 55;
%F Formula
The row polynomials satisfy P(n,x) = (x+(n-2)*x^2)*P(n-2,x) - x^3*d/dx P(n-2,x) + (1+(n-1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(x + (n - 2)*x^2)*P[n - 2, x] - x^3*D[P[n - 2, x], x] + (1 + (n - 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (x + (n - 2)*x^2)*v[i - 2] - x^3*deriv(v[i - 2], x) + (1 + (n - 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], 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

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