Real-rooted atlas

A118933 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-24 - 18\,x + 44\,n + 33\,n x - 24\,n^{2} - 18\,n^{2} x + 4\,n^{3} + 3\,n^{3} x\right)\,P_{n-4}(x) + \left(1\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 1\)

Rows ?

nx0x1Pn(1)
011
111
211
311
4251843
512190211
6361270631
78416301471

All 15 rows

Technical details
Node id
n-279a2b082de9205c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-88a48308c112c49e

Observed on the stored rows ?

  • Degrees: 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3 (irregular)
  • 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:      1;
  3:      1;
  4:     25,     18;
  5:    121,     90;
  6:    361,    270;
  7:    841,    630;
  8:  21841,  31500,  11340;
  9: 184465, 274428, 102060;
%F Formula
The row polynomials satisfy P(n,x) = (4*n^3-24*n^2+44*n-24+(3*n^3-18*n^2+33*n-18)*x)*P(n-4,x) + P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1, P(3,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n^3 - 24*n^2 + 44*n - 24 + (3*n^3 - 18*n^2 + 33*n - 18)*x)*P[n - 4, x] + P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; v[4] = 1; for(n = 4, N, my(i = n + 1); v[i] = (4*n^3 - 24*n^2 + 44*n - 24 + (3*n^3 - 18*n^2 + 33*n - 18)*x)*v[i - 4] + v[i - 1]); 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