Real-rooted atlas

A099093 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(17 + 90\,x + 168\,x^{2} + 134\,x^{3} + 39\,x^{4} - 11\,n - 60\,n x - 126\,n x^{2} - 116\,n x^{3} - 39\,n x^{4}\right)\,P_{n-1}(x) + \left(7 + 55\,x + 162\,x^{2} + 226\,x^{3} + 151\,x^{4} + 39\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13912
2126690168
3453699998912304
41711854748813338882931680
564887484698012538816621287480435456
6245739690265923945540188106319843388667815985792
7931517520314067276248583165791612626805723005053858834982280448

All 15 rows

Technical details
Node id
n-944e86db7073f824 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-60dee4d68f98f7ab

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:      3,       9;
  2:     12,      66,       90;
  3:     45,     369,      999,       891;
  4:    171,    1854,     7488,     13338,      8829;
  5:    648,    8748,    46980,    125388,    166212,      87480;
  6:   2457,   39690,   265923,    945540,   1881063,    1984338,     866781;
  7:   9315,  175203,  1406727,   6248583,  16579161,   26268057,   23005053,    8588349;
  8:  35316,  757998,  7093170,  37791198, 125357706,  265052250,  348769638,  261070938,   85096170;
  9: 133893, 3229227, 34507944, 214415424, 853574922, 2257367454, 3965223456, 4460377752, 2915032617, 843160671;
%F Formula
The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(11*n-17+(60*n-90)*x+(126*n-168)*x^2+(116*n-134)*x^3+(39*n-39)*x^4)*P(n-1,x) + (7+55*x+162*x^2+226*x^3+151*x^4+39*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[(-(11*n - 17 + (60*n - 90)*x + (126*n - 168)*x^2 + (116*n - 134)*x^3 + (39*n - 39)*x^4)*P[n - 1, x] + (7 + 55*x + 162*x^2 + 226*x^3 + 151*x^4 + 39*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*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] = (-(11*n - 17 + (60*n - 90)*x + (126*n - 168)*x^2 + (116*n - 134)*x^3 + (39*n - 39)*x^4)*v[i - 1] + (7 + 55*x + 162*x^2 + 226*x^3 + 151*x^4 + 39*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*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