Real-rooted atlas

A026729 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-4 - 14\,x - 10\,x^{2} + 4\,n + 8\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 12\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
346212
482014345
5165664305171
632144232172608648
764352736760416116132457
8128832214428642168940218219315

All 15 rows

Technical details
Node id
n-90d459f6a295699a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ab1d67dd1b7f53fc

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,    1;
   3:   4,    6,     2;
   4:   8,   20,    14,     3;
   5:  16,   56,    64,    30,     5;
   6:  32,  144,   232,   172,    60,     8;
   7:  64,  352,   736,   760,   416,   116,    13;
   8: 128,  832,  2144,  2864,  2168,   940,   218,   21;
   9: 256, 1920,  5888,  9696,  9456,  5640,  2024,  402,  34;
  10: 512, 4352, 15488, 30400, 36480, 27936, 13736, 4204, 730, 55;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (4*n-4+(8*n-14)*x+(5*n-10)*x^2)*P(n-1,x) - (4*x+12*x^2+5*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[((4*n - 4 + (8*n - 14)*x + (5*n - 10)*x^2)*P[n - 1, x] - (4*x + 12*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n - 2)]];
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] = ((4*n - 4 + (8*n - 14)*x + (5*n - 10)*x^2)*v[i - 1] - (4*x + 12*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n - 2)); 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