Real-rooted atlas

A257626 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 24\,x + 30\,x^{2} + 12\,x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
161218
236180180396
321620524860324010368
41296210608910013608068040315576
57776205092138510036450004082400163296010958328
6466561939140196975807960680015002820013226976044089920427678056
7279936180121322654580601548658440437225040063783417604629441600132269760018535139928

All 15 rows

Technical details
Node id
n-baf71172ebf4970d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cef044b741d89456

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:        6,         12;
  2:       36,        180,         180;
  3:      216,       2052,        4860,         3240;
  4:     1296,      21060,       89100,       136080,         68040;
  5:     7776,     205092,     1385100,      3645000,       4082400,       1632960;
  6:    46656,    1939140,    19697580,     79606800,     150028200,     132269760,       44089920;
  7:   279936,   18012132,   265458060,   1548658440,    4372250400,    6378341760,     4629441600,     1322697600;
  8:  1679616,  165468420,  3455678700,  28008121680,  111222334440,  238879186560,   283321825920,   174596083200,   43649020800;
  9: 10077696, 1509371172, 43950170700, 482324041800, 2590172575200, 7685798944320, 13249461859200, 13213749024000, 7071141369600, 1571364748800;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+24*x+30*x^2+12*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+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[((6 + 24*x + 30*x^2 + 12*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 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] = ((6 + 24*x + 30*x^2 + 12*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 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