Real-rooted atlas

A059473 → γ → basis A013611

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
26814
3163248
44412864236
5120448384952
6328150419205124264
789648648192409618048
82448153603225624576409678736
966884761611980812288040960337952
1018272145536427008552960286720327681463264
11499204398081474560231014416384003932166306048
121363841316864496742491422728273920314572826214427244736
133726083912704164024323470131238305792201850883670016117549952
1410179841155020853274624127426560166297600112852992330301442097152507547264
Technical details
Node id
n-419a8af7baedce0f · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-20fd5d29cb9f346f

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:    6,     8;
  3:   16,    32;
  4:   44,   128,     64;
  5:  120,   448,    384;
  6:  328,  1504,   1920,    512;
  7:  896,  4864,   8192,   4096;
  8: 2448, 15360,  32256,  24576,  4096;
  9: 6688, 47616, 119808, 122880, 40960;
%F Formula
The row polynomials satisfy P(n,x) = (2+8*x)*P(n-2,x) + 2*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(2 + 8*x)*P[n - 2, x] + 2*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (2 + 8*x)*v[i - 2] + 2*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

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