Real-rooted atlas

A144440 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 3\,x\right)\,P_{n-2}(x) + \left(-1 - 2\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
21313127
3110165571333
412492498148223315463
5178164454037502117139371112509
6306619919857100348247386987378375312794959
76185828216503982901258618405470547566463322515017181378189

All 15 rows

Technical details
Node id
n-d7b01e320c8e461b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4eb074acc1a49a5d

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:          2,           1;
  2:         13,          13,           1;
  3:        110,         165,          57,           1;
  4:       1249,        2498,        1482,         233,           1;
  5:      17816,       44540,       37502,       11713,         937,          1;
  6:     306619,      919857,     1003482,      473869,       87378,       3753,         1;
  7:    6185828,    21650398,    29012586,    18405470,     5475664,     633225,     15017,        1;
  8:  143193901,   572775604,   910341541,   726310009,   301187155,   60095833,   4518922,    60073,      1;
  9: 3741598910, 16837195095, 30959883060, 29786013600, 15909223716, 4594813845, 639054432, 31977873, 240297, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+3*x)*P(n-2,x) + (3*n-1+(3*n-2)*x)*P(n-1,x) - (3*x+3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 2, x] + (3*n - 1 + (3*n - 2)*x)*P[n - 1, x] - (3*x + 3*x^2)*D[P[n - 1, x], 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 + x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 2] + (3*n - 1 + (3*n - 2)*x)*v[i - 1] - (3*x + 3*x^2)*deriv(v[i - 1], x)); 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