Real-rooted atlas

A131689 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −1/2]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5 + 26\,x + 49\,x^{2} + 40\,x^{3} + 12\,x^{4} - 2\,n - 11\,n x - 22\,n x^{2} - 19\,n x^{3} - 6\,n x^{4}\right)\,P_{n-1}(x) + \left(2 + 13\,x + 33\,x^{2} + 41\,x^{3} + 25\,x^{4} + 6\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
33111024
4137012574282
575533141516637304416
65414776168142950625809900286448
74683494032166125052586612474603711332102030784
8472935803223045363885944615430939160914549302341229975455656912

All 15 rows

Technical details
Node id
n-a9043801c591f282 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-267f746fabfee8f1

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 -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:       1;
   2:       1,         2;
   3:       3,        11,        10;
   4:      13,        70,       125,         74;
   5:      75,       533,      1415,       1663,        730;
   6:     541,      4776,     16814,      29506,      25809,        9002;
   7:    4683,     49403,    216612,     505258,     661247,      460371,      133210;
   8:   47293,    580322,   3045363,    8859446,   15430939,    16091454,     9302341,     2299754;
   9:  545835,   7637393,  46666237,  162635541,  353589679,   491082611,   425481807,   210262823,   45375130;
  10: 7087261, 111365500, 776478764, 3152918782, 8216691722, 14252014426, 16453226420, 12190693954, 5260325177, 1007179562;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(2*n-5+(11*n-26)*x+(22*n-49)*x^2+(19*n-40)*x^3+(6*n-12)*x^4)*P(n-1,x) + (2+13*x+33*x^2+41*x^3+25*x^4+6*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 5 + (11*n - 26)*x + (22*n - 49)*x^2 + (19*n - 40)*x^3 + (6*n - 12)*x^4)*P[n - 1, x] + (2 + 13*x + 33*x^2 + 41*x^3 + 25*x^4 + 6*x^5)*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] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n - 5 + (11*n - 26)*x + (22*n - 49)*x^2 + (19*n - 40)*x^3 + (6*n - 12)*x^4)*v[i - 1] + (2 + 13*x + 33*x^2 + 41*x^3 + 25*x^4 + 6*x^5)*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

References

  1. [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