Real-rooted atlas

A256890 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 4 + 20\,x + 20\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
14202044
2876180120384
316260110016808403896
43284457001500016800672045096
56426602702010920020580018144060480586664
612882361213807081201999200291648021168006048008475144
7256252205267004268880169520403640896043182720266112006652800134628776
851276684223290024504600131594400390479040673142400671328000359251200798336002332443336

All 15 rows

Technical details
Node id
n-0b37b712dda89f30 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cef044b741d89456

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:    4,     20,       20;
   2:    8,     76,      180,       120;
   3:   16,    260,     1100,      1680,        840;
   4:   32,    844,     5700,     15000,      16800,        6720;
   5:   64,   2660,    27020,    109200,     205800,      181440,        60480;
   6:  128,   8236,   121380,    708120,    1999200,     2916480,      2116800,       604800;
   7:  256,  25220,   526700,   4268880,   16952040,    36408960,     43182720,     26611200,      6652800;
   8:  512,  76684,  2232900,  24504600,  131594400,   390479040,    673142400,    671328000,    359251200,     79833600;
   9: 1024, 232100,  9315020, 135920400,  961098600,  3786108480,   8899450560,  12773376000,  10977120000,   5189184000,  1037836800;
  10: 2048, 700396, 38420580, 735492120, 6718034400, 34191548160, 105270580800, 203954889600, 250278336000, 188806464000, 79913433600, 14529715200;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+8*x+10*x^2+4*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) = 4 + 20*x + 20*x^2.
%t Mathematica
P[1, x_] := 4 + 20*x + 20*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + 8*x + 10*x^2 + 4*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] = 4 + 20*x + 20*x^2; for(n = 2, N, my(i = n); v[i] = ((2 + 8*x + 10*x^2 + 4*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