Real-rooted atlas

A257606 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(4 + 16\,x + 20\,x^{2} + 8\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2167272160
36448810807202352
425629521087215840792037840
51024168089180021960023760095040661872
6409692232702072245520043164003706560123552012512080
71638449392850425202420712061538400851558406054048017297280254291952

All 15 rows

Technical details
Node id
n-9b354df4384dcaca · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-79e4efb037c60552

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:      4,        8;
  2:     16,       72,        72;
  3:     64,      488,      1080,        720;
  4:    256,     2952,     10872,      15840,       7920;
  5:   1024,    16808,     91800,     219600,     237600,       95040;
  6:   4096,    92232,    702072,    2455200,    4316400,     3706560,     1235520;
  7:  16384,   493928,   5042520,   24207120,   61538400,    85155840,    60540480,    17297280;
  8:  65536,  2600712,  34700472,  219875040,  758585520,  1504863360,  1712430720,  1037836800,   259459200;
  9: 262144, 13527848, 231609240, 1886130000, 8487309600, 22646796480, 36687530880, 35390234880, 18681062400, 4151347200;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (4+16*x+20*x^2+8*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4 + 16*x + 20*x^2 + 8*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, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4 + 16*x + 20*x^2 + 8*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 - 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