Real-rooted atlas

A118931 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 6\,x - 12\,n - 9\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-3}(x) + \left(1\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
111
211
39615
4332457
58160141
680110803602241
747616930252014211
818369272161008055665
919779340370427216060480934137
101568961342792024948006048008096481
1181818011818531013471920332640043165431
129521072126108676026644464011975040019958400762450921
139236221292692337076294115536014270256002594592008243599365
14602906595317869053384198583585209807557760181621440055380250017
Technical details
Node id
n-b1f50e3247c41c7c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-92263ec8eaaaba5e

Observed on the stored rows ?

  • Degrees: 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4 (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:      9,      6;
  4:     33,     24;
  5:     81,     60;
  6:    801,   1080,    360;
  7:   4761,   6930,   2520;
  8:  18369,  27216,  10080;
  9: 197793, 403704, 272160, 60480;
%F Formula
The row polynomials satisfy P(n,x) = (4*n^2-12*n+8+(3*n^2-9*n+6)*x)*P(n-3,x) + P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n^2 - 12*n + 8 + (3*n^2 - 9*n + 6)*x)*P[n - 3, 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; for(n = 3, N, my(i = n + 1); v[i] = (4*n^2 - 12*n + 8 + (3*n^2 - 9*n + 6)*x)*v[i - 3] + 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