Real-rooted atlas

A099040 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(6 + 34\,x + 66\,x^{2} + 54\,x^{3} + 16\,x^{4} - 4\,n - 22\,n x - 48\,n x^{2} - 46\,n x^{3} - 16\,n x^{4}\right)\,P_{n-1}(x) + \left(2 + 18\,x + 58\,x^{2} + 86\,x^{3} + 60\,x^{4} + 16\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
264272120
3161685766401400
44460831086944569616400
5120206414040471367795250688192000
63286744572402564246386888375044510722248000
7896214402181121221952406835280424968732672401408026320000

All 15 rows

Technical details
Node id
n-371e9500eef8f7d7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ce326fbf2bbf31e5

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:    2,      8;
  2:    6,     42,      72;
  3:   16,    168,     576,      640;
  4:   44,    608,    3108,     6944,      5696;
  5:  120,   2064,   14040,    47136,     77952,     50688;
  6:  328,   6744,   57240,   256424,    638688,    837504,    451072;
  7:  896,  21440,  218112,  1221952,   4068352,   8042496,   8732672,    4014080;
  8: 2448,  66816,  792288,  5328000,  22211856,  58745088,  96192000,   89100288,  35721216;
  9: 6688, 205056, 2776896, 21790848, 109148832, 361729152, 792781824, 1107431424, 894246912, 317882368;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(4*n-6+(22*n-34)*x+(48*n-66)*x^2+(46*n-54)*x^3+(16*n-16)*x^4)*P(n-1,x) + (2+18*x+58*x^2+86*x^3+60*x^4+16*x^5)*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*n - 6 + (22*n - 34)*x + (48*n - 66)*x^2 + (46*n - 54)*x^3 + (16*n - 16)*x^4)*P[n - 1, x] + (2 + 18*x + 58*x^2 + 86*x^3 + 60*x^4 + 16*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*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*n - 6 + (22*n - 34)*x + (48*n - 66)*x^2 + (46*n - 54)*x^3 + (16*n - 16)*x^4)*v[i - 1] + (2 + 18*x + 58*x^2 + 86*x^3 + 60*x^4 + 16*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*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

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741