Real-rooted atlas

A046739 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 2\,x + n + n x\right)\,P_{n-2}(x) + \left(2 + x + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 9 + 9\,x + x^{2}\)
\(P_{2}(x) = 44 + 66\,x + 24\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
199119
24466241135
32655303205511171
41854463539401275118111823
51483344499488252348545712451136459
61334964672366285444032701233681540050011771815
713349615339844851037668416742885694598416499141011125561891
814684570660805651217595601177824906363058818808230275232015778520341405658143

All 15 rows

Technical details
Node id
n-f85d6b4ccee7473c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-fe1bba5e6d9fef5e

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          9,           9,           1;
   2:         44,          66,          24,           1;
   3:        265,         530,         320,          55,           1;
   4:       1854,        4635,        3940,        1275,         118,           1;
   5:      14833,       44499,       48825,       23485,        4571,         245,          1;
   6:     133496,      467236,      628544,      403270,      123368,       15400,        500,         1;
   7:    1334961,     5339844,     8510376,     6841674,     2885694,      598416,      49914,      1011,        1;
   8:   14684570,    66080565,   121759560,   117782490,    63630588,    18808230,    2752320,    157785,     2034,       1;
   9:  176214841,   881074205,  1842913545,  2085208950,  1376892594,   532958118,  114934380,  12200595,   490589,    4081,    1;
  10: 2290792932, 12599361126, 29493959000, 38227607055, 29873769288, 14358220800, 4146081588, 670596135, 52682564, 1508430, 8176, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+2+(n+2)*x)*P(n-2,x) + (n+2+(n+1)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 9 + 9*x + x^2, P(2,x) = 44 + 66*x + 24*x^2 + x^3.
%t Mathematica
P[1, x_] := 9 + 9*x + x^2;
P[2, x_] := 44 + 66*x + 24*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(n + 2 + (n + 2)*x)*P[n - 2, x] + (n + 2 + (n + 1)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 9 + 9*x + x^2; v[2] = 44 + 66*x + 24*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = (n + 2 + (n + 2)*x)*v[i - 2] + (n + 2 + (n + 1)*x)*v[i - 1] - (x + 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, v[i])))};
T(10)

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