Real-rooted atlas

A128099 → basis A013611 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n\right)\,P_{n}(x) = \left(-80 - 64\,x + 84\,n + 64\,n x\right)\,P_{n-1}(x) + \left(-205 - 324\,x - 128\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
211819
3211637
413118464379
53415121921045
61651340023045127867
75061112488320204826677

All 15 rows

Technical details
Node id
n-81b00881f7f71483 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c1122197787ef9b0

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:     1;
  2:    11,      8;
  3:    21,     16;
  4:   131,    184,     64;
  5:   341,    512,    192;
  6:  1651,   3400,   2304,    512;
  7:  5061,  11248,   8320,   2048;
  8: 21571,  58456,  58560,  25600,  4096;
  9: 72181, 211424, 231744, 112640, 20480;
%F Formula
The row polynomials satisfy 4*n*P(n,x) = (84*n-80+(64*n-64)*x)*P(n-1,x) - (205+324*x+128*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[Together[((84*n - 80 + (64*n - 64)*x)*P[n - 1, x] - (205 + 324*x + 128*x^2)*D[P[n - 1, x], x])/(4*n)]];
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] = ((84*n - 80 + (64*n - 64)*x)*v[i - 1] - (205 + 324*x + 128*x^2)*deriv(v[i - 1], x))/(4*n)); 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 ?

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