Real-rooted atlas

A099040 → basis A013609 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 8\,x\right)\,P_{n-2}(x) + \left(6 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16814
24210464210
3288105612805123136
4198096001734413824409646844
5136088208019699223500813926432768699720
693528674784201945632076802850816134348826214410451896
7642816539827219353600383877124548198432178176125829122097152156122624

All 15 rows

Technical details
Node id
n-2e5dc2f7cc79a54b · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-69a27ac83c6caaaa

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:        6,         8;
  2:       42,       104,         64;
  3:      288,      1056,       1280,        512;
  4:     1980,      9600,      17344,      13824,       4096;
  5:    13608,     82080,     196992,     235008,     139264,      32768;
  6:    93528,    674784,    2019456,    3207680,    2850816,    1343488,     262144;
  7:   642816,   5398272,   19353600,   38387712,   45481984,   32178176,   12582912,    2097152;
  8:  4418064,  42329088,  176822784,  420556800,  622759936,  587792384,  345243648,  115343360,   16777216;
  9: 30365280, 326851200, 1558877184, 4323078144, 7681007616, 9065758720, 7106723840, 3567255552, 1040187392, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (6+8*x)*P(n-2,x) + (6+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 8*x;
P[n_, x_] := P[n, x] = Expand[(6 + 8*x)*P[n - 2, x] + (6 + 8*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] = 6 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (6 + 8*x)*v[i - 2] + (6 + 8*x)*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

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
  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