Real-rooted atlas

A063967 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2 - 10\,x - 12\,x^{2} + 2\,n + 6\,n x + 6\,n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 8\,x^{2} - 6\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3410620
4832401696
5168817214444464
6322246007764881202240
764544185632723160159232810816
812812805312119041561612032505689652224

All 15 rows

Technical details
Node id
n-6898798eb3977109 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-75d07e1be1e68206

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:   2,    2;
   3:   4,   10,     6;
   4:   8,   32,    40,     16;
   5:  16,   88,   172,    144,     44;
   6:  32,  224,   600,    776,    488,    120;
   7:  64,  544,  1856,   3272,   3160,   1592,    328;
   8: 128, 1280,  5312,  11904,  15616,  12032,   5056,    896;
   9: 256, 2944, 14400,  39232,  65296,  68160,  43680,  15744,  2448;
  10: 512, 6656, 37504, 120448, 243488, 321952, 278976, 153024, 48288, 6688;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(6*n-10)*x+(6*n-12)*x^2)*P(n-1,x) - (2*x+8*x^2+6*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (6*n - 10)*x + (6*n - 12)*x^2)*P[n - 1, x] - (2*x + 8*x^2 + 6*x^3)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n - 2 + (6*n - 10)*x + (6*n - 12)*x^2)*v[i - 1] - (2*x + 8*x^2 + 6*x^3)*deriv(v[i - 1], x))/(n - 1)); 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