Real-rooted atlas

A257608 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-7 - 8\,x + 9\,n + 9\,n x\right)\,P_{n-1}(x) + \left(-9\,x - 9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
22222145
344066022211323
41276025520149822222155485
5484880121220010142003091002222213042603
62278936068368080742632003457960061173422222221206339805
7127620416044667145605966933280375054680010905756401186739402222222116671870603

All 15 rows

Technical details
Node id
n-97fcf61accf79726 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c5a9a6448f2c45fc

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:             2,              1;
  2:            22,             22,              1;
  3:           440,            660,            222,              1;
  4:         12760,          25520,          14982,           2222,              1;
  5:        484880,        1212200,        1014200,         309100,          22222,             1;
  6:      22789360,       68368080,       74263200,       34579600,        6117342,        222222,            1;
  7:    1276204160,     4466714560,     5966933280,     3750546800,     1090575640,     118673940,      2222222,           1;
  8:   82953270400,   331813081600,   526115164960,   416999709280,   170396925160,   32909596720,   2279249302,    22222222,         1;
  9: 6138542009600, 27623439043200, 50698486460160, 48535320408960, 25657069782960, 7259064535800, 967053694200, 43550181180, 222222222, 1;
%F Formula
The row polynomials satisfy P(n,x) = (9*n-7+(9*n-8)*x)*P(n-1,x) - (9*x+9*x^2)*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[(9*n - 7 + (9*n - 8)*x)*P[n - 1, x] - (9*x + 9*x^2)*D[P[n - 1, x], x]];
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 - 7 + (9*n - 8)*x)*v[i - 1] - (9*x + 9*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 - 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