Real-rooted atlas

A046739 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4} + n x + 3\,n x^{2} + 3\,n x^{3} + n x^{4}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 9\,x + 9\,x^{2}\)
\(P_{2}(x) = 1 + 24\,x + 66\,x^{2} + 44\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
119919
21246644135
31553205302651171
41118127539404635185411823
51245457123485488254449914833136459
61500154001233684032706285444672361334961771815
711011499145984162885694684167485103765339844133496125561891
812034157785275232018808230636305881177824901217595606608056514684570405658143

All 15 rows

Technical details
Node id
n-012b9166110551b5 · revision 0 · JSON
Definition source
derived:mobius_x_over_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: 1,    9,       9;
   2: 1,   24,      66,       44;
   3: 1,   55,     320,      530,       265;
   4: 1,  118,    1275,     3940,      4635,       1854;
   5: 1,  245,    4571,    23485,     48825,      44499,       14833;
   6: 1,  500,   15400,   123368,    403270,     628544,      467236,      133496;
   7: 1, 1011,   49914,   598416,   2885694,    6841674,     8510376,     5339844,     1334961;
   8: 1, 2034,  157785,  2752320,  18808230,   63630588,   117782490,   121759560,    66080565,    14684570;
   9: 1, 4081,  490589, 12200595, 114934380,  532958118,  1376892594,  2085208950,  1842913545,   881074205,   176214841;
  10: 1, 8176, 1508430, 52682564, 670596135, 4146081588, 14358220800, 29873769288, 38227607055, 29493959000, 12599361126, 2290792932;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = ((n+2)*x+(3*n+6)*x^2+(3*n+6)*x^3+(n+2)*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 9*x + 9*x^2, P(2,x) = 1 + 24*x + 66*x^2 + 44*x^3.
%t Mathematica
P[1, x_] := 1 + 9*x + 9*x^2;
P[2, x_] := 1 + 24*x + 66*x^2 + 44*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(((n + 2)*x + (3*n + 6)*x^2 + (3*n + 6)*x^3 + (n + 2)*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 9*x + 9*x^2; v[2] = 1 + 24*x + 66*x^2 + 44*x^3; for(n = 3, N, my(i = n); v[i] = (((n + 2)*x + (3*n + 6)*x^2 + (3*n + 6)*x^3 + (n + 2)*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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