Real-rooted atlas

A166341 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 12\,x + 12\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
11121225
21267248147
31542944802401069
4111010443096360014409291
5122234621656033480302401008094045
614461105280088250200382320282240806401086987
7189434494364560165144037951204651920290304072576014127229
811790106164159621610080000326793605912928060439680326592007257600203949291

All 15 rows

Technical details
Node id
n-83613b7858f852b2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b6466966904f5921

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,   12,     12;
   2: 1,   26,     72,       48;
   3: 1,   54,    294,      480,       240;
   4: 1,  110,   1044,     3096,      3600,       1440;
   5: 1,  222,   3462,    16560,     33480,      30240,      10080;
   6: 1,  446,  11052,    80088,    250200,     382320,     282240,       80640;
   7: 1,  894,  34494,   364560,   1651440,    3795120,    4651920,     2903040,      725760;
   8: 1, 1790, 106164,  1596216,  10080000,   32679360,   59129280,    60439680,    32659200,     7257600;
   9: 1, 3582, 323862,  6809520,  58381080,  256556160,  642660480,   956551680,   837889920,   399168000,   79833600;
  10: 1, 7166, 982332, 28533528, 325953000, 1889623440, 6294516480, 12793697280, 16149974400, 12370579200, 5269017600, 958003200;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,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 >= 2, with P(1,x) = 1 + 12*x + 12*x^2.
%t Mathematica
P[1, x_] := 1 + 12*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((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 + 12*x + 12*x^2; for(n = 2, N, my(i = n); v[i] = ((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