Real-rooted atlas

A209599 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(4 + 4\,n + n^{2}\right)\,P_{n}(x) = \left(26 - 4\,x - 19\,n + 2\,n x + 11\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(144 + 50\,x - 8\,x^{2} - 72\,n - 41\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(108 + 102\,x + 24\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
2415
38311
4169126
53223459
66457151137
7128135465314
8256313133221725
95127113607861667
10102415939392553013842
1120483527237077112078843
12409677375841222043539120369
138192168391410861321446173846898
14163843640933527164134536686491108005
Technical details
Node id
n-97af8fbb64a585df · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-27d55a3d589e9c6a

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:   2;
  2:   4,   1;
  3:   8,   3;
  4:  16,   9,   1;
  5:  32,  23,   4;
  6:  64,  57,  15,  1;
  7: 128, 135,  46,  5;
  8: 256, 313, 133, 22, 1;
  9: 512, 711, 360, 78, 6;
%F Formula
The row polynomials satisfy (n^2+4*n+4)*P(n,x) = (11*n^2-19*n+26+(2*n^2+2*n-4)*x)*P(n-1,x) - (72*n-144+(41*n-50)*x+(4*n+8)*x^2)*d/dx P(n-1,x) + (108+102*x+24*x^2)*d^2/dx^2 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[((11*n^2 - 19*n + 26 + (2*n^2 + 2*n - 4)*x)*P[n - 1, x] - (72*n - 144 + (41*n - 50)*x + (4*n + 8)*x^2)*D[P[n - 1, x], x] + (108 + 102*x + 24*x^2)*D[P[n - 1, x], {x, 2}])/(n^2 + 4*n + 4)]];
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] = ((11*n^2 - 19*n + 26 + (2*n^2 + 2*n - 4)*x)*v[i - 1] - (72*n - 144 + (41*n - 50)*x + (4*n + 8)*x^2)*deriv(v[i - 1], x) + (108 + 102*x + 24*x^2)*deriv(deriv(v[i - 1], x), x))/(n^2 + 4*n + 4)); 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

Candidate equalities ?

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