Real-rooted atlas

A059473 → x ↦ x+1 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 4\,x\right)\,P_{n-2}(x) + \left(6 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2405216108
3264512320641160
4174444964240172825612464
51152037056467522886487041024133920
6760962933764636803840641756164198440961438912
750265622589444290816446156827402249932801966081638415460480

All 15 rows

Technical details
Node id
n-ea60ffc5610195a1 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-e1118c67b24d5944

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,         4;
  2:       40,        52,        16;
  3:      264,       512,       320,        64;
  4:     1744,      4496,      4240,      1728,       256;
  5:    11520,     37056,     46752,     28864,      8704,      1024;
  6:    76096,    293376,    463680,    384064,    175616,     41984,      4096;
  7:   502656,   2258944,   4290816,   4461568,   2740224,    993280,    196608,    16384;
  8:  3320320,  17042176,  37808896,  47323648,  36526336,  17790976,   5337088,   901120,   65536;
  9: 21932544, 126580736, 321221120, 470187008, 437259776, 267785216, 107945984, 27607040, 4063232, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (4+4*x)*P(n-2,x) + (6+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[(4 + 4*x)*P[n - 2, x] + (6 + 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] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 4*x)*v[i - 2] + (6 + 4*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